# Diagonal double Kodaira fibrations with minimal signature

Francesco Polizzi  Dipartimento di Matematica e Informatica
Università della Calabria
Ponte Pietro Bucci 30B, I-87036 Arcavacata di Rende, Cosenza, Italy
<EMAIL_ADDRESS>and  Pietro Sabatino  Via Val Sillaro 5
00141 Roma, Italy<EMAIL_ADDRESS>
###### Abstract.

We study some special systems of generators on finite groups, introduced in
previous work by the first author and called _diagonal double Kodaira
structures_ , in order to investigate non-abelian, finite quotients of the
pure braid group on two strands $\mathsf{P}_{2}(\Sigma_{b})$, where
$\Sigma_{b}$ is a closed Riemann surface of genus $b$. In particular, we prove
that, if a finite group $G$ admits a diagonal double Kodaira structure, then
$|G|\geq 32$, and equality holds if and only if $G$ is extra-special. In the
last section, as a geometrical application of our algebraic results, we
construct two $3$-dimensional families of double Kodaira fibrations having
signature $16$.

###### Key words and phrases:

Surface braid groups, extra-special $p$-groups, Kodaira fibrations

_2010 Mathematics Subject Classification._ 14J29, 14J25, 20D15

###### Contents

  1. 0 Introduction
  2. 1 Group-theoretical preliminaries: CCT-groups and extra-special groups
  3. 2 Diagonal double Kodaira structures
  4. 3 Structures on groups of order at most $32$
    1. 3.1 Prestructures
    2. 3.2 The case $|G|<32$
    3. 3.3 The case $|G|=32$ and $G$ non-extra-special
    4. 3.4 The case $|G|=32$ and $G$ extra-special
  5. 4 Geometrical application: diagonal double Kodaira fibrations
    1. 4.1 The computation of $H_{1}(S,\,\mathbb{Z})$

##  0\. Introduction

A _Kodaira fibration_ is a smooth, connected holomorphic fibration
$f_{1}\colon S\longrightarrow B_{1}$, where $S$ is a compact complex surface
and $B_{1}$ is a compact closed curve, which is not isotrivial (this means
that not all fibres are biholomorphic each other). The genus $b_{1}:=g(B_{1})$
is called the _base genus_ of the fibration, and the genus $g:=g(F)$, where
$F$ is any fibre, is called the _fibre genus_. A surface $S$ that is the total
space of a Kodaira fibration is called a _Kodaira fibred surface_. For every
Kodaira fibration, we have $b_{1}\geq 2$ and $g\geq 3$, see [Kas68, Theorem
1.1]. Since the fibration is smooth, the condition on the base genus implies
that $S$ contains no rational or elliptic curves; hence $S$ is minimal and, by
the sub-additivity of the Kodaira dimension, it is of general type, hence
algebraic.

An important topological invariant of a Kodaira fibred surface $S$ is its
_signature_ $\sigma(S)$, namely the signature of the intersection form on the
middle cohomology group $H^{2}(S,\,\mathbb{R})$. Actually, the first examples
of Kodaira fibrations (see [Kod67]) were constructed in order to show that
$\sigma$ is not multiplicative for fibre bundles. In fact, $\sigma(S)>0$ for
every Kodaira fibration (see the introduction to [LLR20]), whereas
$\sigma(B_{1})=\sigma(F)=0$, hence $\sigma(S)\neq\sigma(B_{1})\sigma(F)$; by
[CHS57], this in turn means that the monodromy action of $\pi_{1}(B)$ on the
rational cohomology ring $H^{*}(S,\,\mathbb{Q})$ is non-trivial.

Every Kodaira fibred surface $S$ has the structure of a real surface bundle
over a smooth real surface, and so $\sigma(S)$ is divisible by $4$, see
[Mey73]. If, in addition, $S$ has a spin structure, i.e. its canonical class
is $2$-divisible in $\operatorname{Pic}(S)$, then $\sigma(S)$ is a positive
multiple of $16$ by Rokhlin’s theorem, and examples with $\sigma(S)=16$ are
constructed in [LLR20]. It is not known whether there exists a Kodaira fibred
surface with $\sigma(S)\leq 12$.

Kodaira fibred surfaces are a source of fascinating ad deep questions at the
cross-road between the algebro-geometric properties of a compact, complex
surface and the topological properties of the underlying closed, oriented
$4$-manifold. In fact, they can be studied by using, besides the usual
algebro-geometric methods, techniques borrowed from geometric topology such as
the Meyer signature formula, the Birman-Hilden relations in the mapping class
group and the subtraction of Lefchetz fibrations, see [En98, EKKOS02, St02,
L17]. We refer the reader to the survey paper [Cat17] and the references
contained therein for further details.

The original example by Kodaira, and its variants described in [At69, Hir69],
are obtained by taking ramified covers of products of curves, so they come
with a pair of Kodaira fibrations. This leads to the definition of “double”
Kodaira fibration, see [Zaal95, LeBrun00, BDS01, BD02, CatRol09, Rol10,
LLR20]:

######  Definition 0.1.

A _double Kodaira surface_ is a compact, complex surface $S$, endowed with a
_double Kodaira fibration_ , namely a surjective, holomorphic map $f\colon
S\longrightarrow B_{1}\times B_{2}$ yielding, by composition with the natural
projections, two Kodaira fibrations $f_{i}\colon S\longrightarrow B_{i}$,
$i=1,\,2$.

In the sequel, we will describe our approach to the construction of double
Kodaira fibrations based on the techniques introduced in [CaPol19, Pol20], and
present our results. The main step is to “detopologize” the problem, by
transforming it into a purely algebraic one. This will be done in the
particular case of _diagonal_ double Kodaira fibrations, namely, Stein
factorizations of finite Galois covers

(1) $\mathbf{f}\colon S\longrightarrow\Sigma_{b}\times\Sigma_{b},$

branched with order $n\geq 2$ over the diagonal
$\Delta\subset\Sigma_{b}\times\Sigma_{b}$, where $\Sigma_{b}$ is a closed
Riemann surface of genus $b$. By Grauert-Remmert’s extension theorem and
Serre’s GAGA, the existence of a $G$-cover $\mathbf{f}$ as in (1), up to cover
isomorphisms, is equivalent to the existence of a group epimorphism

(2) $\varphi\colon\pi_{1}(\Sigma_{b}\times\Sigma_{b}-\Delta)\longrightarrow
G,$

up to automorphisms of $G$. Furthermore, the condition that $\mathbf{f}$ is
branched of order $n$ over $\Delta$ is rephrased by asking that
$\varphi(\gamma_{\Delta})$ has order $n$ in $G$, where $\gamma_{\Delta}$ is
the homotopy class in $\Sigma_{b}\times\Sigma_{b}-\Delta$ of a loop in
$\Sigma_{b}\times\Sigma_{b}$ that “winds once” around $\Delta$. The
requirement $n\geq 2$ means that $\varphi$ does not factor through
$\pi_{1}(\Sigma_{b}\times\Sigma_{b})$; it also implies that $G$ is non-
abelian, because $\gamma_{\Delta}$ is a non-trivial commutator in
$\pi_{1}(\Sigma_{b}\times\Sigma_{b}-\Delta)$.

Recall now that the group $\pi_{1}(\Sigma_{b}\times\Sigma_{b}-\Delta)$ is
isomorphic to $\mathsf{P}_{2}(\Sigma_{b})$, the pure braid group of genus $b$
on two strands; such a group admit a geometric presentation with $4g+1$
generators

(3) $\rho_{11},\,\tau_{11},\ldots,\rho_{1b},\,\tau_{1b},\,A_{12},$

where $A_{12}$ corresponds to $\gamma_{\Delta}$, subject to the set of
relations written in Section 2, see [GG04, Theorem 7]. Taking the images of
these generators via the group epimorphism $\eqref{eq:intr-varphi}$, we get an
ordered set

(4)
$\mathfrak{S}=(\mathsf{r}_{11},\,\mathsf{t}_{11},\ldots,\mathsf{r}_{1b},\,\mathsf{t}_{1b},\,\mathsf{r}_{21},\,\mathsf{t}_{21},\ldots,\mathsf{r}_{2b},\,\mathsf{t}_{2b},\,\mathsf{z})$

of $4b+1$ generators of $G$, such that $o(\mathsf{z})=n$. This will be called
a _diagonal double Kodaira structure_ of type $(b,\,n)$ on $G$, see Definition
2.1. In the light of the previous considerations, we see that the geometric
problem of constructing a $G$-cover $\mathbf{f}$ as in (1) is now translated
into the combinatorial-algebraic problem of finding a diagonal double Kodaira
structure of type $(b,\,n)$ in $G$.

It turns out that the $G$-cover $\mathbf{f}$ is a diagonal double Kodaira
fibration (namely, the two surjective maps $f_{i}\colon
S\longrightarrow\Sigma_{b}$, obtained as composition with the natural
projections, have connected fibres) if and only if $\mathfrak{S}$ is _strong_
, an additional condition introduced in Definition 2.8; furthermore, the
algebraic signature $\sigma(\mathfrak{S})$, see Definition 2.7, equals the
geometric signature $\sigma(S)$.

Summing up, classifying diagonal double Kodaira fibrations is equivalent to
describing finite groups which admit a diagonal double Kodaira structure. Our
first main result in this direction is the following:

######  Theorem A (see Proposition 3.9, 3.11 and Theorem 3.15).

Let $G$ be a finite group admitting a diagonal double Kodaira structure. Then
$|G|\geq 32$, with equality if and only if $G$ is extra-special $($see
$\operatorname{Section}$ $\operatorname{\ref{sec:CCT}}$ for the definition$)$.
Moreover, the following holds.

  * $\boldsymbol{(1)}$

Both extra-special groups $G$ of order $32$ admit $2211840=1152\cdot 1920$
diagonal double Kodaira structures of type $(b,\,n)=(2,\,2)$. Every such a
structure $\mathfrak{S}$ is strong and satisfies $\sigma(\mathfrak{S})=16$.

  * $\boldsymbol{(2)}$

If $G=G(32,\,49)=\mathsf{H}_{5}(\mathbb{Z}_{2})$, these structures form $1920$
orbits under the action of $\operatorname{Aut}(G)$.

  * $\boldsymbol{(3)}$

If $G=G(32,\,50)=\mathsf{G}_{5}(\mathbb{Z}_{2})$, these structures form $1152$
orbits under the action of $\operatorname{Aut}(G)$.

Theorem A should be compared with previous results, obtained by the first
author in collaboration with A. Causin, regarding the construction of diagonal
double Kodaira structures on some extra-special groups of order at least
$2^{7}=128$, see [CaPol19, Pol20]. It turns out that the examples presented
here are really new, in the sense that they cannot be obtained as images of
structures on extra-special groups of larger order, see Remark 3.17.

A restatement of Theorem A in terms of surface braid groups is the following,
cf. Remark 3.18. First of all, let us say that a quotient map/group
epimorphism
$\varphi\colon\pi_{1}(\Sigma_{b}\times\Sigma_{b}-\Delta)\longrightarrow G$ is
_admissible_ if $\varphi(A_{12})$ has order $n\geq 2$, then:

######  Theorem A’.

Let $G$ be a finite group admitting an admissible epimorphism
$\varphi\colon\mathsf{P}_{2}(\Sigma_{b})\longrightarrow G$. Then $|G|\geq 32$,
with equality if and only if $G$ is extra-special. Moreover, the following
holds.

  * $\boldsymbol{(1)}$

For both extra-special groups $G$ of order $32$, there are $2211840=1152\cdot
1920$ admissible epimorphisms
$\varphi\colon\mathsf{P}_{2}(\Sigma_{2})\longrightarrow G$. For all of them,
$\varphi(A_{12})$ is the generator of $Z(G)$, so $n=2$.

  * $\boldsymbol{(2)}$

If $G=G(32,\,49)=\mathsf{H}_{5}(\mathbb{Z}_{2})$, these epimorhisms form
$1920$ orbits under the natural action of $\operatorname{Aut}(G)$.

  * $\boldsymbol{(3)}$

If $G=G(32,\,50)=\mathsf{G}_{5}(\mathbb{Z}_{2})$, these epimorhisms form
$1152$ orbits under the natural action of $\operatorname{Aut}(G)$.

The geometrical counterpart of Theorems A and A’ can be now expressed in terms
of diagonal double Kodaira fibrations as follows:

######  Theorem B (see Theorem 4.7.).

Let $G$ be a finite group and $\mathbf{f}\colon
S\longrightarrow\Sigma_{b}\times\Sigma_{b}$ be a Galois cover, with Galois
group $G$, branched on the diagonal $\Delta$ with branching order $n\geq 2$.
Then $|G|\geq 32$, with equality if and only if $G$ is extra-special.
Moreover, the following holds.

  * $\boldsymbol{(1)}$

For both extra-special groups of order $32$, there exist $2211840=1152\cdot
1920$ distinct $G$-covers $\mathbf{f}\colon
S\longrightarrow\Sigma_{2}\times\Sigma_{2}$ as above. All of them are diagonal
double Kodaira fibrations with $n=2$ and

(5) $b_{1}=b_{2}=2,\quad g_{1}=g_{2}=41,\quad\sigma(S)=16.$

  * $\boldsymbol{(2)}$

If $G=G(32,\,49)=\mathsf{H}_{5}(\mathbb{Z}_{2})$, these $G$-covers form $1920$
equivalence classes up to cover isomorphisms.

  * $\boldsymbol{(3)}$

If $G=G(32,\,50)=\mathsf{G}_{5}(\mathbb{Z}_{2})$, these $G$-covers form $1152$
equivalence classes up to cover isomorphisms.

As a consequence, we obtain a sharp lower bound for the signature of a
diagonal double Kodaira fibration or, equivalently, of a diagonal double
Kodaira structure:

######  Theorem C (see Corollary 4.8).

Let $f\colon S\longrightarrow\Sigma_{b_{1}}\times\Sigma_{b_{2}}$ be a diagonal
double Kodaira fibration, associated with a diagonal double Kodaira structure
of type $(b,\,n)$ on a finite group $G$. Then $\sigma(S)\geq 16$, and equality
holds precisely when $(b,\,n)=(2,\,2)$ and $G$ is an extra-special group of
order $32$.

These results yield, as a by-product, new “double solutions” to a problem
(stated by G. Mess) from Kirby’s problem list in low-dimensional topology
[Kir97, Problem 2.18 A], asking what is the smallest number $b$ for which
there exists a real surface bundle over a real surface with base genus $b$ and
non-zero signature. We actually have $b=2$, also for double Kodaira
fibrations, as shown in [CaPol19, Proposition 3.19] and [Pol20, Theorem 3.6]
by using double Kodaira structures of type $(2,\,3)$ on extra-special groups
of order $3^{5}$. Those fibrations had signature $144$ and fibre genera $325$;
we are now able to substantially lower both these values:

######  Theorem D (see Theorem 4.9).

Let $S$ be a diagonal double Kodaira surface, associated with a strong
diagonal double Kodaira structure of type $(b,\,n)=(2,\,2)$ on an extra-
special group $G$ of order $32$. Then the real manifold $M$ underlying $S$ is
a closed, orientable $4$-manifold of signature $16$ that can be realized as a
real surface bundle over a real surface of genus $2$, with fibre genus $41$,
in two different ways.

In fact, we may ask whether $16$ and $41$ are the minimum possible values for
the signature and the fibre genus of a (non necessarily diagonal) double
Kodaira fibration $f\colon S\longrightarrow\Sigma_{2}\times\Sigma_{2}$, cf.
Corollary 4.10.

We believe that the results described above are significant for at least two
reasons:

  * $\boldsymbol{(i)}$

although we know that $\mathsf{P}_{2}(\Sigma_{b})$ is residually $p$-finite
for all prime number $p\geq 2$, see [BarBel09, pp. 1481-1490], so far there
has been no systematic work aimed to describe its admissible quotients. The
first results in this direction were those of A. Causin and the first author,
who showed that both extra-special groups of order $p^{4b+1}$ appear as
admissible quotients of $\mathsf{P}_{2}(\Sigma_{b})$ for all $b\geq 2$ and all
prime numbers $p\geq 5$; moreover, if $p$ divides $b+1$, then both extra-
special groups of order $p^{2b+1}$ appear as admissible quotients, too. As we
said before, the smallest adimissible quotients detected in [CaPol19] and
[Pol20], corresponding to the case $(b,\,p)=(3,\,2)$, have order $2^{7}=128$.

Our Theorem B sheds some new light on this problem, by providing a sharp lower
bound for the order of $G$: more precisely, if a finite group $G$ is an
admissible quotient of $\mathsf{P}_{2}(\Sigma_{b})$ for some $b$, then
$|G|\geq 32$, with equality if and only if $G$ is extra-special. Moreover, for
both extra-special groups of order $32$, Theorem B computes the number of
admissible quotient maps
$\varphi\colon\mathsf{P}_{2}(\Sigma_{2})\longrightarrow G$, and the number of
their equivalence classes up to the natural action of $\operatorname{Aut}(G)$;

  * $\boldsymbol{(ii)}$

constructing (double) Kodaira fibrations with small signature is a rather
difficult problem. As far as we know, before the present work the only
examples with signature $16$ were the ones listed in [LLR20, Table 3, Cases
6.2, 6.6, 6.7 (Type 1), 6.9]. Our examples in Theorem A are new, since both
the base genera and the fibre genera are different from the ones in the
aforementioned cases. Note that our results also show that _every_ curve of
genus $2$ (and not only some special curve with extra automorphisms) is the
base of a double Kodaira fibration with signature $16$. Thus, we obtain two
families of dimension $3$ of such fibrations that, to the best of our
knowledge, provide the first examples of positive-dimensional families of
double Kodaira fibrations with small signature.

Finally, this work also contain a Computer Algebra part, concerning the
calculation of the group $H_{1}(S,\,\mathbb{Z})$, where $S$ is as in Theorem
D, by using the software `GAP4`, see [GAP4]. The result is the following:

######  Theorem E (see Proposition 4.14).

Let $f\colon S\longrightarrow\Sigma_{2}\times\Sigma_{2}$ be the diagonal
double Kodaira fibration associated with a diagonal double Kodaira structure
of type $(b,\,n)=(2,\,2)$ on an extra-special group $G$ of order $32$. Then

(6) $H_{1}(S,\,\mathbb{Z})=\mathbb{Z}^{8}\oplus(\mathbb{Z}_{2})^{4}.$

In particular, this homology group is independent both on $G$ and on the
chosen structure on it.

Thus $\mathsf{b}_{1}(S)=8$ and, subsequently, the pull-back map $f^{*}\colon
H^{1}(\Sigma_{b}\times\Sigma_{b},\,\mathbb{Q})\longrightarrow
H^{1}(S,\,\mathbb{Q})$ is an isomorphism. Following [Breg18], we will express
this fact by saying that $f$ is _maximal_ , see Proposition 4.18. For an
interpretation of maximality in terms of monodromy, see Corollary 4.17.

Let us now describe how this paper is organized. In Section 1 we introduce
some algebraic preliminaries, in particular we discuss the so-called CCT-
groups (Definition 1.1), namely, finite non-abelian groups in which
commutativity is a transitive relation on the set of non-central elements.
These groups are of historical importance in the context of classification of
finite simple groups, see Remark 1.3, and they play a fundamental role in this
paper, as we will soon explain. It turns out that there are precisely eight
groups $G$ with $|G|\leq 32$ that are not CCT-groups, namely $\mathsf{S}_{4}$
and seven groups of order $32$, see Corollary 1.6, Proposition 1.7 and
Proposition 1.14.

In Section 2 we define diagonal double Kodaira structures on finite groups and
we explain the relation with their counterpart in geometric topology, namely
admissible group epimorphisms from pure surface braid groups.

Section 3 is devoted to the study of diagonal double Kodaira structures in
groups of order at most $32$. One crucial technical result is Proposition 3.4,
stating that there are no such structures on CCT-groups. Thus, in order to
prove the first part of Theorem A, we only need to exclude the existence of
diagonal double Kodaira structures on $\mathsf{S}_{4}$ and on the five non-
abelian, non-CCT groups of order $32$; this is done in Proposition 3.9 and
Proposition 3.11, respectively. The second part of Theorem A, i.e. the
computation of number of structures in each case, is obtained by using some
techniques borrowed from [Win72]; more precisely, we exploit the fact that
$V=G/Z(G)$ is a symplectic vector space of dimension $4$ over
$\mathbb{Z}_{2}$, and that $\operatorname{Out}(G)$ embeds in
$\mathrm{Sp}(4,\,\mathbb{Z}_{2})$ as the orthogonal group associated with the
quadratic form $q\colon V\longrightarrow\mathbb{Z}_{2}$ related to the
symplectic form $(\cdot\;,\cdot)$ by
$q(\overline{\mathsf{x}}\,\overline{\mathsf{y}})=q(\overline{\mathsf{x}})+q(\overline{\mathsf{y}})+(\overline{\mathsf{x}},\,\overline{\mathsf{y}})$.

Finally, in Section 4 we establish the relation between our algebraic results
and the geometrical framework of diagonal double Kodaira fibrations, and we
provide the proofs of Theorems B, C, and D; furthermore, we state Theorem E,
discussing some of its consequences.

The paper ends with two appendices. In Appendix A we collect the presentations
for the non-abelian groups of order $24$ and $32$ that we used in our
calculations, while Appendix B contains the details about the computational
proof of Theorem E.

$\mathbf{Notation\;and\;conventions}$. If $S$ is a complex, non-singular
projective surface, then $c_{1}(S)$, $c_{2}(S)$ denote the first and second
Chern class of its tangent bundle $T_{S}$, respectively. If $X$ is a
topological space, the fundamental group of $X$ will be denoted by
$\pi_{1}(X)$ and its first Betti number by $\mathsf{b}_{1}(X)$.

Throughout the paper we use the following notation for groups:

  * •

$\mathbb{Z}_{n}$: cyclic group of order $n$.

  * •

$G=N\rtimes Q$: semi-direct product of $N$ and $Q$, namely, split extension of
$Q$ by $N$, where $N$ is normal in $G$.

  * •

$G=N.Q$: non-split extension of $Q$ by $N$.

  * •

$\operatorname{Aut}(G)$: the automorphism group of $G$.

  * •

$\mathsf{D}_{p,\,q,\,r}=\mathbb{Z}_{q}\rtimes\mathbb{Z}_{p}=\langle
x,\,y\;|\;x^{p}=y^{q}=1,\;xyx^{-1}=y^{r}\rangle$: split metacyclic group of
order $pq$. The group $\mathsf{D}_{2,\,n,\,-1}$ is the dihedral group of order
$2n$ and will be denoted by $\mathsf{D}_{2n}$.

  * •

If $n$ is an integer greater or equal to $4$, we denote by
$\mathsf{QD}_{2^{n}}$ the quasi-dihedral group of order $2^{n}$, having
presentation

(7) $\mathsf{QD}_{2^{n}}:=\langle x,\,y\mid
x^{2}=y^{2^{n-1}}=1,\;xyx^{-1}=y^{2^{n-2}-1}\rangle.$

  * •

The generalized quaternion group of order $4n$ is denoted by $\mathsf{Q}_{4n}$
and is presented as

(8) $\mathsf{Q}_{4n}=\langle x,\,y,\,z\mid x^{n}=y^{2}=z^{2}=xyz\rangle.$

For $n=2$ we obtain the usual quaternion group $\mathsf{Q}_{8}$, for which we
adopt the classical presentation

(9) $\mathsf{Q}_{8}=\langle i,\,j,\,k\mid i^{2}=j^{2}=k^{2}=ijk\rangle,$

denoting by $-1$ the unique element of order $2$.

  * •

$\mathsf{S}_{n},\;\mathsf{A}_{n}$: symmetric, alternating group on $n$
symbols. We write the composition of permutations from the right to the left;
for instance, $(13)(12)=(123)$.

  * •

$\mathsf{GL}(n,\,\mathbb{F}_{q}),\,\mathsf{SL}(n,\,\mathbb{F}_{q}),\,\mathsf{Sp}(n,\,\mathbb{F}_{q})$:
general linear group, special linear group and symplectic group of $n\times n$
matrices over a field with $q$ elements.

  * •

The order of a finite group $G$ is denoted by $|G|$. If $x\in G$, the order of
$x$ is denoted by $o(x)$ and its centralizer in $G$ by $C_{G}(x)$.

  * •

If $x,\,y\in G$, their commutator is defined as $[x,\,y]=xyx^{-1}y^{-1}$.

  * •

The commutator subgroup of $G$ is denoted by $[G,\,G]$, the center of $G$ by
$Z(G)$.

  * •

If $S=\\{s_{1},\ldots,s_{n}\\}\subset G$, the subgroup generated by $S$ is
denoted by $\langle S\rangle=\langle s_{1},\ldots,s_{n}\rangle$.

  * •

$\mathrm{IdSmallGroup}(G)$ indicates the label of the group $G$ in the `GAP4`
database of small groups. For instance
$\mathrm{IdSmallGroup}(\mathsf{D}_{4})=G(8,\,3)$ means that $\mathsf{D}_{4}$
is the third in the list of groups of order $8$.

  * •

If $N$ is a normal subgroup of $G$ and $g\in G$, we denote by $\bar{g}$ the
image of $g$ in the quotient group $G/N$.

##  1\. Group-theoretical preliminaries: CCT-groups and extra-special groups

######  Definition 1.1.

A non-abelian, finite group $G$ is said to be a _center commutative-transitive
group_ $($or a CCT-_group_ , for short$)$ if commutativity is a transitive
relation on the set on non-central elements of $G$. In other words, if
$x,\,y,\,z\in G-Z(G)$ and $[x,\,y]=[y,\,z]=1$, then $[x,\,z]=1$.

######  Proposition 1.2.

For a finite group $G$, the following properties are equivalent.

  * $\boldsymbol{(1)}$

$G$ is a _CCT_ -group.

  * $\boldsymbol{(2)}$

For every pair $x,\,y$ of non-central elements in $G$, the relation
$[x,\,y]=1$ implies $C_{G}(x)=C_{G}(y)$.

  * $\boldsymbol{(3)}$

For every non-central element $x\in G$, the centralizer $C_{G}(x)$ is abelian.

###### Proof.

$\boldsymbol{(1)\Rightarrow(2)}$ Take two commuting elements $x,\,y\in G-Z(G)$
and let $z\in C_{G}(x)$. If $z$ is central then $z\in C_{G}(y)$ by definition,
otherwise $[x,\,y]=[x,\,z]=1$ implies $[y,\,z]=1$ by the assumption that $G$
is a CCT-group. Therefore we get $C_{G}(x)\subseteq C_{G}(y)$, and exchanging
the roles of $x,\,y$ we can deduce the reverse inclusion.

$\boldsymbol{(2)\Rightarrow(3)}$ Given any element $x\in G-Z(G)$, it is
sufficient to check that $[y,\,z]=1$ for every pair of non-central elements
$y,\,z\in C_{G}(x)$. By assumption, $C_{G}(y)=C_{G}(z)$, hence $y\in C_{G}(z)$
and we are done.

$\boldsymbol{(3)\Rightarrow(1)}$ Let $x,\,y,\,z\in G-Z(G)$ and suppose
$[x,\,y]=[y,\,z]=1$, namely, $x,\,z\in C_{G}(y)$. Since we are assuming that
$C_{G}(y)$ is abelian, this gives $[x,\,z]=1$, hence $G$ is a CCT-group. ∎

######  Remark 1.3.

CCT-groups are of historical importance in the context of classification of
finite simple groups, see for instance [Suz61], where they are called CA-
_groups_. Further references on the topic are [Schm70], [Reb71], [Rocke73],
[Wu98].

######  Lemma 1.4.

If $G$ is a finite group such that $G/Z(G)$ is cyclic, then $G$ is abelian.

###### Proof.

Every element of $G$ can be written as $zy^{n}$, where $y\in G$ is such that
its image generates $G/Z(G)$, $z\in Z(G)$ and $n\in\mathbb{Z}$. It follows
that any two elements of $G$ commute. ∎

######  Proposition 1.5.

Let $G$ be a non-abelian, finite group.

  * $\boldsymbol{(1)}$

If $|G|$ is the product of at most three prime factors $($non necessarily
distinct$)$, then $G$ is a _CCT_ -group.

  * $\boldsymbol{(2)}$

If $|G|=p^{4}$, with $p$ prime, then $G$ is a _CCT_ -group.

  * $\boldsymbol{(3)}$

If $G$ contains an abelian normal subgroup of prime index, then $G$ is a _CCT_
-group.

###### Proof.

$\boldsymbol{(1)}$ Assume that $|G|$ is the product of at most three prime
factors, and take a non-central element $y$. Then the centralizer $C_{G}(y)$
has non-trivial center, because $1\neq y\in C_{G}(y)$, and its order is the
product of at most two primes. Therefore the quotient of $C_{G}(y)$ by its
center is cyclic, hence $C_{G}(y)$ is abelian by Lemma 1.4.

$\boldsymbol{(2)}$ Assume $|G|=p^{4}$ and suppose by contradiction that there
exist three elements $x,\,y,\,z\in G-Z(G)$ such that $[x,\,y]=[y,\,z]=1$ but
$[x,\,z]\neq 1$. They generate a non-abelian subgroup $N=\langle
x,\,y,\,z\rangle$, which is not the whole of $G$ since $y\in Z(N)$ but
$y\notin Z(G)$. It follows that $N$ has order $p^{3}$ and so, by Lemma 1.4,
its center is cyclic of order $p$, generated by $y$. The group $G$ is a finite
$p$-group, hence a nilpotent group; being a proper subgroup of maximal order
in a nilpotent group, $N$ is normal in $G$ (see [Mac12, Corollary 5.2]), so we
have a conjugacy homomorphism $G\longrightarrow\mathrm{Aut}(N)$, that in turn
induces a conjugacy homomorphism
$G\longrightarrow\mathrm{Aut}(Z(N))\simeq\mathbb{Z}_{p-1}$. The image of such
a homomorphism must have order dividing both $p^{4}$ and $p-1$, hence it is
trivial. In other words, the conjugacy action of $G$ on $Z(N)=\langle
y\rangle$ is trivial, hence $y$ is central in $G$, contradiction.

$\boldsymbol{(3)}$ Let $N$ be an abelian normal subgroup of $G$ such that
$G/N$ has prime order $p$. As $G/N$ has no non-trivial proper subgroups, it
follows that $N$ is a maximal subgroup of $G$. Let $x$ be any non-central
element of $G$, so that $C_{G}(x)$ is a proper subgroup of $G$; then there are
two possibilities.

Case 1: $x\in N$. Then $N\subseteq C_{G}(x)$ and so, by the maximality of $N$,
we get $C_{G}(x)=N$, which is abelian.
Case 2: $x\notin N$. Then the image of $x$ generates $G/N$, and so every
element $y\in G$ can be written in the form $y=ux^{r}$, where $u\in N$ and
$0\leq r\leq p-1$. In particular, if $y\in C_{G}(x)$, the condition
$[x,\,y]=1$ yields $[x,\,u]=1$, namely $u\in N\cap C_{G}(x)$. Since $N$ is
abelian, it follows that $C_{G}(x)$ is abelian, too.

∎

We now want to classify non-abelian, non-CCT groups of order at most $32$.
First of all, as an immediate consequence of Parts $\boldsymbol{(1)}$ and
$\boldsymbol{(2)}$ of Proposition 1.5, we have the following

######  Corollary 1.6.

Let $G$ be a non-abelian, finite group such that $|G|\leq 32$. If $G$ is not a
_CCT_ -group, then either $|G|=24$ or $|G|=32$.

Let us start by disposing of the case $G=24$.

######  Proposition 1.7.

Let $G$ be a non-abelian finite group such that $|G|=24$ and $G$ is not a
_CCT_ -group. Then $G=\mathsf{S}_{4}$.

###### Proof.

We start by observing that $\mathsf{S}_{4}$ is not a CCT-group. In fact,
$(1234)$ commutes to its square $(13)(24)$, which commutes to $(12)(34)$, but
$(1234)$ and $(12)(34)$ do not commute.

What is left is to show that the remaining non-abelian groups of order $24$
are all CCT-groups; we will do a case-by-case analysis, referring the reader
to the presentations given in Table 1 of Appendix A. Apart from
$G=G(24,\,3)=\mathsf{SL}(2,\,\mathbb{F}_{3})$, for which we give an ad-hoc
proof, we will show that all these groups contain an abelian subgroup $N$ of
prime index, so that we can conclude by using Part $\boldsymbol{(3)}$ of
Proposition 1.5.

  * •

$G=G(24,\,1).$ Take $N=\langle x^{2}y\rangle\simeq\mathbb{Z}_{12}$.

  * •

$G=G(24,\,3).$ The action of $\mathrm{Aut}(G)$ has five orbits, whose
representative elements are $\\{1,\,x,\,x^{2},\,z,\,z^{2}\\}$, see [SL(2,3)].
We have $\langle z^{2}\rangle=Z(G)$ and so, since $C_{G}(x)\subseteq
C_{G}(x^{2})$, it suffices to show that the centralizers of $x^{2}$ and $z$
are both abelian. In fact, we have

(10) $C_{G}(x^{2})=\langle x\rangle\simeq\mathbb{Z}_{6},\quad C_{G}(z)=\langle
z\rangle\simeq\mathbb{Z}_{4}.$

  * •

$G=G(24,\,4).$ Take $N=\langle x\rangle\simeq\mathbb{Z}_{12}$.

  * •

$G=G(24,\,5).$ Take $N=\langle y\rangle\simeq\mathbb{Z}_{12}$.

  * •

$G=G(24,\,6).$ Take $N=\langle y\rangle\simeq\mathbb{Z}_{12}$.

  * •

$G=G(24,\,7).$ Take $N=\langle
z,\,x^{2}y\rangle\simeq\mathbb{Z}_{6}\times\mathbb{Z}_{2}$.

  * •

$G=G(24,\,8).$ Take $N=\langle
y,\,z,\,w\rangle\simeq\mathbb{Z}_{6}\times\mathbb{Z}_{2}$.

  * •

$G=G(24,\,10).$ Take $N=\langle z,\,y\rangle\simeq\mathbb{Z}_{12}$.

  * •

$G=G(24,\,11).$ Take $N=\langle z,\,i\rangle\simeq\mathbb{Z}_{12}$.

  * •

$G=G(24,\,13).$ Take $N=\langle
z\rangle\times\mathsf{V}_{4}\simeq(\mathbb{Z}_{2})^{3}$, where
$\mathsf{V}_{4}=\langle(1\,2)(3\,4),\;(1\,3)(2\,4)\rangle$ is the Klein
subgroup.

  * •

$G=G(24,\,14).$ Take $N=\langle
z,\,w\rangle\times\langle(123)\rangle\simeq\mathbb{Z}_{6}\times\mathbb{Z}_{2}$.

This completes the proof. ∎

The next step is to classify non-abelian, non-CCT groups $G$ with $|G|=32$; it
will turn out that there are precisely seven of them, see Proposition 1.14.
Before doing this, let us introduce the following classical definition, see
for instance [Gor07, p. 183] and [Is08, p. 123].

######  Definition 1.8.

Let $p$ be a prime number. A finite $p$-group $G$ is called _extra-special_ if
its center $Z(G)$ is cyclic of order $p$ and the quotient $V=G/Z(G)$ is a non-
trivial, elementary abelian $p$-group.

An elementary abelian $p$-group is a finite-dimensional vector space over the
field $\mathbb{Z}_{p}$, hence it is of the form $V=(\mathbb{Z}_{p})^{\dim V}$
and $G$ fits into a short exact sequence

(11) $1\longrightarrow\mathbb{Z}_{p}\longrightarrow G\longrightarrow
V\longrightarrow 1.$

Note that, $V$ being abelian, we must have $[G,\,G]=\mathbb{Z}_{p}$, namely
the commutator subgroup of $G$ coincides with its center. Furthermore, since
the extension (11) is central, it cannot be split, otherwise $G$ would be
isomorphic to the direct product of the two abelian groups $\mathbb{Z}_{p}$
and $V$, which is impossible because $G$ is non-abelian.

If $G$ is extra-special, then we can define a map $\omega\colon V\times
V\longrightarrow\mathbb{Z}_{p}$ as follows: for every $v_{1},\,v_{2}\in V$, we
set $\omega(v_{1},\,v_{2})=[g_{1},\,g_{2}]$, where $g_{i}$ is any lift of
$v_{i}$ in $G$. This turns out to be a symplectic form on $V$, hence $\dim V$
is even and $|G|=p^{\dim V+1}$ is an odd power of $p$.

For every prime number $p$, there are precisely two isomorphism classes
$M(p)$, $N(p)$ of non-abelian groups of order $p^{3}$, namely

$\begin{split}M(p)&=\langle\mathsf{r},\,\mathsf{t},\,\mathsf{z}\;|\;\mathsf{r}^{p}=\mathsf{t}^{p}=1,\,\mathsf{z}^{p}=1,[\mathsf{r},\,\mathsf{z}]=[\mathsf{t},\,\mathsf{z}]=1,\,[\mathsf{r},\,\mathsf{t}]=\mathsf{z}^{-1}\rangle\\\
N(p)&=\langle\mathsf{r},\,\mathsf{t},\,\mathsf{z}\;|\;\mathsf{r}^{p}=\mathsf{t}^{p}=\mathsf{z},\,\mathsf{z}^{p}=1,[\mathsf{r},\,\mathsf{z}]=[\mathsf{t},\,\mathsf{z}]=1,\,[\mathsf{r},\,\mathsf{t}]=\mathsf{z}^{-1}\rangle\\\
\end{split}$

and both of them are in fact extra-special, see [Gor07, Theorem 5.1 of Chapter
5].

If $p$ is odd, then the groups $M(p)$ and $N(p)$ are distinguished by their
exponent, which equals $p$ and $p^{2}$, respectively. If $p=2$, the group
$M(p)$ is isomorphic to the dihedral group $D_{8}$, whereas $N(p)$ is
isomorphic to the quaternion group $\mathsf{Q}_{8}$.

The classification of extra-special $p$-groups is now provided by the result
below, see [Gor07, Section 5 of Chapter 5] and [CaPol19, Section 2].

######  Proposition 1.9.

If $b\geq 2$ is a positive integer and $p$ is a prime number, there are
exactly two isomorphism classes of extra-special $p$-groups of order
$p^{2b+1}$, that can be described as follows.

  * •

The central product $\mathsf{H}_{2b+1}(\mathbb{Z}_{p})$ of $b$ copies of
$M(p)$, having presentation

(12)
$\begin{split}\mathsf{H}_{2b+1}(\mathbb{Z}_{p})=\langle\,&\mathsf{r}_{1},\,\mathsf{t}_{1},\ldots,\mathsf{r}_{b},\,\mathsf{t}_{b},\,\mathsf{z}\;|\;\mathsf{r}_{j}^{p}=\mathsf{t}_{j}^{p}=\mathsf{z}^{p}=1,\\\
&[\mathsf{r}_{j},\,\mathsf{z}]=[\mathsf{t}_{j},\,\mathsf{z}]=1,\\\
&[\mathsf{r}_{j},\,\mathsf{r}_{k}]=[\mathsf{t}_{j},\,\mathsf{t}_{k}]=1,\\\
&[\mathsf{r}_{j},\,\mathsf{t}_{k}]=\mathsf{z}^{-\delta_{jk}}\,\rangle.\end{split}$

If $p$ is odd, this group has exponent $p$ and is isomorphic to the matrix
Heisenberg group
$\mathcal{H}_{2b+1}(\mathbb{Z}_{p})\subset\mathsf{GL}(b+2,\,\mathbb{Z}_{p})$
of dimension $2b+1$ over the field $\mathbb{Z}_{p}$.

  * •

The central product $\mathsf{G}_{2b+1}(\mathbb{Z}_{p})$ of $b-1$ copies of
$M(p)$ and one copy of $N(p)$, having presentation

(13)
$\begin{split}\mathsf{G}_{2b+1}(\mathbb{Z}_{p})=\langle\,&\mathsf{r}_{1},\,\mathsf{t}_{1},\ldots,\mathsf{r}_{b},\,\mathsf{t}_{b},\,\mathsf{z}\;|\;\mathsf{r}_{b}^{p}=\mathsf{t}_{b}^{p}=\mathsf{z},\\\
&\mathsf{r}_{1}^{p}=\mathsf{t}_{1}^{p}=\ldots=\mathsf{r}_{b-1}^{p}=\mathsf{t}_{b-1}^{p}=\mathsf{z}^{p}=1,\\\
&[\mathsf{r}_{j},\,\mathsf{z}]=[\mathsf{t}_{j},\,\mathsf{z}]=1,\\\
&[\mathsf{r}_{j},\,\mathsf{r}_{k}]=[\mathsf{t}_{j},\,\mathsf{t}_{k}]=1,\\\
&[\mathsf{r}_{j},\,\mathsf{t}_{k}]=\mathsf{z}^{-\delta_{jk}}\,\rangle.\end{split}$

If $p$ is odd, this group has exponent $p^{2}$.

######  Remark 1.10.

In both cases, from the relations above we deduce

(14)
$[\mathsf{r}_{j}^{-1},\,\mathsf{t}_{k}]=\mathsf{z}^{\delta_{jk}},\quad[\mathsf{r}_{j}^{-1},\,\mathsf{t}_{k}^{-1}]=\mathsf{z}^{-\delta_{jk}}$

######  Remark 1.11.

For both groups $\mathsf{H}_{2b+1}(\mathbb{Z}_{p})$ and
$\mathsf{G}_{2b+1}(\mathbb{Z}_{p})$, the center coincides with the derived
subgroup and is equal to $\langle\mathsf{z}\rangle\simeq\mathbb{Z}_{p}$.

######  Remark 1.12.

If $p=2$, we can distinguish the two groups
$\mathsf{H}_{2b+1}(\mathbb{Z}_{p})$ and $\mathsf{G}_{2b+1}(\mathbb{Z}_{p})$ by
counting the number of elements of order $4$.

######  Remark 1.13.

The groups $\mathsf{H}_{2b+1}(\mathbb{Z}_{p})$ and
$\mathsf{G}_{2b+1}(\mathbb{Z}_{p})$ are not CCT-groups. In fact, let us take
two distinct indices $j,\,k\in\\{1,\ldots,b\\}$ and consider the non-central
elements $\mathsf{r}_{j}$, $\mathsf{t}_{j}$, $\mathsf{t}_{k}$. Then we have
$[\mathsf{r}_{j},\,\mathsf{t}_{k}]=[\mathsf{t}_{k},\,\mathsf{t}_{j}]=1$, but
$[\mathsf{r}_{j},\,\mathsf{t}_{j}]=\mathsf{z}^{-1}$.

We can now dispose of the case $|G|=32$.

######  Proposition 1.14.

Let $G$ be a non-abelian, finite group such that $|G|=32$ and $G$ is not a
_CCT_ -group. Then $G=G(32,\,t)$, where
$t\in\\{6,\,7,\,8,\,43,\,44,\,49,\,50\\}$. Here
$G(32,\,49)=\mathsf{H}_{5}(\mathbb{Z}_{2})$ and
$G(32,\,50)=\mathsf{G}_{5}(\mathbb{Z}_{2})$ are the two extra-special groups
of order $32$, in particular they have nilpotence class $2$, whereas the
remaining five groups have nilpotence class $3$.

###### Proof.

We first do a case-by case analysis showing that, if
$t\notin\\{6,\,7,\,8,\,43,\,44,\,49,\,50\\}$, then $G=G(32,\,t)$ contains an
abelian subgroup $N$ of index $2$, so that $G$ is a CCT-group by Part
$\boldsymbol{(3)}$ of Proposition 1.5. In every case, we refer the reader to
the presentation given in Table 2 of Appendix A.

  * •

$G=G(32,\,2).$ Take $N=\langle
x,\,y^{2},z\rangle\simeq\mathbb{Z}_{4}\times(\mathbb{Z}_{2})^{2}$.

  * •

$G=G(32,\,4).$ Take $N=\langle x,\,y^{2}\rangle\simeq(\mathbb{Z}_{4})^{2}$.

  * •

$G=G(32,\,5).$ Take $N=\langle
x,\,y\rangle\simeq\mathbb{Z}_{8}\times\mathbb{Z}_{2}$.

  * •

$G=G(32,\,9).$ Take $N=\langle
x,\,y\rangle\simeq\mathbb{Z}_{8}\times\mathbb{Z}_{2}$.

  * •

$G=G(32,\,10).$ Take $N=\langle
ix,\,k\rangle\simeq\mathbb{Z}_{8}\times\mathbb{Z}_{2}$.

  * •

$G=G(32,\,11).$ Take $N=\langle x,\,y\rangle\simeq(\mathbb{Z}_{4})^{2}$.

  * •

$G=G(32,\,12).$ Take $N=\langle x^{2},\,y\rangle\simeq(\mathbb{Z}_{4})^{2}$.

  * •

$G=G(32,\,13).$ Take $N=\langle
x^{2},\,y\rangle\simeq\mathbb{Z}_{8}\times\mathbb{Z}_{2}$.

  * •

$G=G(32,\,14).$ Take $N=\langle
x^{2},\,y\rangle\simeq\mathbb{Z}_{8}\times\mathbb{Z}_{2}$.

  * •

$G=G(32,\,15).$ Take $N=\langle
x^{2},\,y\rangle\simeq\mathbb{Z}_{8}\times\mathbb{Z}_{2}$.

  * •

$G=G(32,\,17).$ Take $N=\langle y\rangle\simeq\mathbb{Z}_{16}$.

  * •

$G=G(32,\,18).$ Take $N=\langle y\rangle\simeq\mathbb{Z}_{16}$.

  * •

$G=G(32,\,19).$ Take $N=\langle y\rangle\simeq\mathbb{Z}_{16}$.

  * •

$G=G(32,\,20).$ Take $N=\langle x\rangle\simeq\mathbb{Z}_{16}$.

  * •

$G=G(32,\,22).$ Take $N=\langle w\rangle\times\langle
x,\,y\rangle\simeq\mathbb{Z}_{8}\times(\mathbb{Z}_{2})^{2}$.

  * •

$G=G(32,\,23).$ Take $N=\langle z\rangle\times\langle
x,\,y^{2}\rangle\simeq\mathbb{Z}_{4}\times(\mathbb{Z}_{2})^{2}$.

  * •

$G=G(32,\,24).$ Take $N=\langle x,\,y\rangle\simeq(\mathbb{Z}_{4})^{2}$.

  * •

$G=G(32,\,25).$ Take $N=\langle z\rangle\times\langle
y^{2}\rangle\simeq(\mathbb{Z}_{4})^{2}$.

  * •

$G=G(32,\,26).$ Take $N=\langle z\rangle\times\langle
i\rangle\simeq(\mathbb{Z}_{4})^{2}$.

  * •

$G=G(32,\,27).$ Take $N=\langle
x,\,y,\,a,\,b\rangle\simeq(\mathbb{Z}_{2})^{4}$.

  * •

$G=G(32,\,28).$ Take $N=\langle
x,\,y,\,z\rangle\simeq\mathbb{Z}_{4}\times(\mathbb{Z}_{2})^{2}$.

  * •

$G=G(32,\,29).$ Take $N=\langle
x,\,i,\,z\rangle\simeq\mathbb{Z}_{4}\times(\mathbb{Z}_{2})^{2}$.

  * •

$G=G(32,\,30).$ Take $N=\langle
x,\,y,\,z\rangle\simeq\mathbb{Z}_{4}\times(\mathbb{Z}_{2})^{2}$.

  * •

$G=G(32,\,31).$ Take $N=\langle x,\,y\rangle\simeq(\mathbb{Z}_{4})^{2}$.

  * •

$G=G(32,\,32).$ Take $N=\langle y,\,z\rangle\simeq(\mathbb{Z}_{4})^{2}$.

  * •

$G=G(32,\,33).$ Take $N=\langle x,\,y\rangle\simeq(\mathbb{Z}_{4})^{2}$.

  * •

$G=G(32,\,34).$ Take $N=\langle x,\,y\rangle\simeq(\mathbb{Z}_{4})^{2}$.

  * •

$G=G(32,\,35).$ Take $N=\langle x,\,k\rangle\simeq(\mathbb{Z}_{4})^{2}$.

  * •

$G=G(32,\,37).$ Take $N=\langle
x,\,y\rangle\simeq\mathbb{Z}_{8}\times\mathbb{Z}_{2}$.

  * •

$G=G(32,\,38).$ Take $N=\langle
x,\,y\rangle\simeq\mathbb{Z}_{8}\times\mathbb{Z}_{2}$.

  * •

$G=G(32,\,39).$ Take $N=\langle z\rangle\times\langle
y\rangle\simeq\mathbb{Z}_{8}\times\mathbb{Z}_{2}$.

  * •

$G=G(32,\,40).$ Take $N=\langle z\rangle\times\langle
y\rangle\simeq\mathbb{Z}_{8}\times\mathbb{Z}_{2}$.

  * •

$G=G(32,\,41).$ Take $N=\langle w\rangle\times\langle
x\rangle\simeq\mathbb{Z}_{8}\times\mathbb{Z}_{2}$.

  * •

$G=G(32,\,42).$ Take $N=\langle
x,\,y\rangle\simeq\mathbb{Z}_{8}\times\mathbb{Z}_{2}$.

  * •

$G=G(32,\,46).$ Take $N=\langle z,\,w\rangle\times\langle
y\rangle\simeq\mathbb{Z}_{4}\times(\mathbb{Z}_{2})^{2}$.

  * •

$G=G(32,\,47).$ Take $N=\langle z,\,w\rangle\times\langle
i\rangle\simeq\mathbb{Z}_{4}\times(\mathbb{Z}_{2})^{2}$.

  * •

$G=G(32,\,48).$ Take $N=\langle
x,\,y,\,z\rangle\simeq\mathbb{Z}_{4}\times(\mathbb{Z}_{2})^{2}$.

It remains to show that $G=G(32,\,t)$ is not a CCT-group for
$t\in\\{6,\,7,\,8,\,43,\,44,\,49,\,50\\}$, and to compute the nilpotency class
in each case. In the sequel, we will denote by

$G=\Gamma_{1}\supseteq\Gamma_{2}\supseteq\Gamma_{3}\supseteq\ldots$

the lower central series of $G$. Recall that a group $G$ has nilpotency class
$c$ if $\Gamma_{c}\neq\\{1\\}$ and $\Gamma_{c+1}=\\{1\\}$.

For $t=49$ and $t=50$ we have the two extra-special cases, that are not CCT-
groups by Remark 1.13; their nilpotency class is $2$ by Remark 1.11. Let us
now deal with the remaining cases. For each of them, we exhibit three non-
central elements for which commutativity is not a transitive relation, and we
show that $\Gamma_{3}\neq\\{1\\}$ and $\Gamma_{4}=\\{1\\}$; note that this
means that $\Gamma_{2}=[G,\,G]$ is not contained in $Z(G)$, whereas
$\Gamma_{3}=[\Gamma_{2},\,G]$ is contained in $Z(G)$.

  * •

$G=G(32,\,6).$ The center of $G$ is $Z(G)=\langle
x\rangle\simeq\mathbb{Z}_{2}$. We have $[y,\,w^{2}]=[w^{2},\,w]=1$, but
$[y,\,w]=x$. The derived subgroup of $G$ is $\Gamma_{2}=[G,\,G]=\langle
x,\,y\rangle\simeq(\mathbb{Z}_{2})^{2}$, and a short computation gives
$\Gamma_{3}=Z(G)$, so $c=3$.

  * •

$G=G(32,\,7).$ The center of $G$ is $Z(G)=\langle
w\rangle\simeq\mathbb{Z}_{2}$. We have $[y,\,z]=[z,\,u]=1$, but $[y,\,u]=w$.
The derived subgroup of $G$ is $\Gamma_{2}=[G,\,G]=\langle
w,\,z\rangle\simeq(\mathbb{Z}_{2})^{2}$, and a short computation gives
$\Gamma_{3}=Z(G)$, so $c=3$.

  * •

$G=G(32,\,8).$ The center of $G$ is $Z(G)=\langle
x^{4}\rangle\simeq\mathbb{Z}_{2}$. We have $[x,\,x^{2}]=[x^{2},\,y]=1$, but
$[x,\,y]=z^{2}$. The derived subgroup of $G$ is $\Gamma_{2}=[G,\,G]=\langle
x^{4},\,y\rangle\simeq(\mathbb{Z}_{2})^{2}$, and a short computation gives
$\Gamma_{3}=Z(G)$, so $c=3$.

  * •

$G=G(32,\,43).$ The center of $G$ is $Z(G)=\langle
x^{4}\rangle\simeq\mathbb{Z}_{2}$. We have $[x,\,x^{2}]=[x^{2},\,z]=1$, but
$[x,\,z]=x^{4}$. The derived subgroup of $G$ is $\Gamma_{2}=[G,\,G]=\langle
x^{2}\rangle\simeq\mathbb{Z}_{4}$, and a short computation gives
$\Gamma_{3}=Z(G)$, so $c=3$.

  * •

$G=G(32,\,44).$ The center of $G$ is $Z(G)=\langle
i^{2}\rangle\simeq\mathbb{Z}_{2}$. We have $[x,\,xk]=[xk,\,z]=1$, but
$[x,\,z]=i^{2}$. The derived subgroup of $G$ is $\Gamma_{2}=[G,\,G]=\langle
k\rangle\simeq\mathbb{Z}_{4}$, and a short computation gives
$\Gamma_{3}=Z(G)$, so $c=3$.

This completes the proof. ∎

##  2\. Diagonal double Kodaira structures

For more details on the material contained in this section, we refer the
reader to [CaPol19] and [Pol20]. Let $G$ be a finite group and let $b,\,n\geq
2$ be two positive integers.

######  Definition 2.1.

A _diagonal double Kodaira structure_ of type $(b,\,n)$ on $G$ is an ordered
set of $4b+1$ generators

(15)
$\mathfrak{S}=(\mathsf{r}_{11},\,\mathsf{t}_{11},\ldots,\mathsf{r}_{1b},\,\mathsf{t}_{1b},\;\mathsf{r}_{21},\,\mathsf{t}_{21},\ldots,\mathsf{r}_{2b},\,\mathsf{t}_{2b},\;\mathsf{z}),$

with $o(\mathsf{z})=n$, such that the following relations are satisfied. We
systematically use the commutator notation in order to indicate relations of
conjugacy type, writing for instance $[x,\,y]=zy^{-1}$ instead of
$xyx^{-1}=z$.

  * •

Surface relations

(16)
$\displaystyle[\mathsf{r}_{1b}^{-1},\,\mathsf{t}_{1b}^{-1}]\,\mathsf{t}_{1b}^{-1}\,[\mathsf{r}_{1\,b-1}^{-1},\,\mathsf{t}_{1\,b-1}^{-1}]\,\mathsf{t}_{1\,b-1}^{-1}\cdots[\mathsf{r}_{11}^{-1},\,\mathsf{t}_{11}^{-1}]\,\mathsf{t}_{11}^{-1}\,(\mathsf{t}_{11}\,\mathsf{t}_{12}\cdots\mathsf{t}_{1b})=\mathsf{z}$
(17)
$\displaystyle[\mathsf{r}_{21}^{-1},\,\mathsf{t}_{21}]\,\mathsf{t}_{21}\,[\mathsf{r}_{22}^{-1},\,\mathsf{t}_{22}]\,\mathsf{t}_{22}\cdots[\mathsf{r}_{2b}^{-1},\,\mathsf{t}_{2b}]\,\mathsf{t}_{2b}\,(\mathsf{t}_{2b}^{-1}\,\mathsf{t}_{2\,b-1}^{-1}\cdots\mathsf{t}_{21}^{-1})=\mathsf{z}^{-1}$

  * •

Conjugacy action of $\mathsf{r}_{1j}$

(18) $\displaystyle[\mathsf{r}_{1j},\,\mathsf{r}_{2k}]$ $\displaystyle=1$
$\displaystyle\mathrm{if}\;\;j<k$ (19)
$\displaystyle[\mathsf{r}_{1j},\,\mathsf{r}_{2j}]$ $\displaystyle=1$ (20)
$\displaystyle[\mathsf{r}_{1j},\,\mathsf{r}_{2k}]$
$\displaystyle=\mathsf{z}^{-1}\,\mathsf{r}_{2k}\,\mathsf{r}_{2j}^{-1}\,\mathsf{z}\,\mathsf{r}_{2j}\,\mathsf{r}_{2k}^{-1}\;\;$
$\displaystyle\mathrm{if}\;\;j>k$ (22)
$\displaystyle[\mathsf{r}_{1j},\,\mathsf{t}_{2k}]$ $\displaystyle=1$
$\displaystyle\mathrm{if}\;\;j<k$ (23)
$\displaystyle[\mathsf{r}_{1j},\,\mathsf{t}_{2j}]$
$\displaystyle=\mathsf{z}^{-1}$ (24)
$\displaystyle[\mathsf{r}_{1j},\,\mathsf{t}_{2k}]$
$\displaystyle=[\mathsf{z}^{-1},\,\mathsf{t}_{2k}]$
$\displaystyle\mathrm{if}\;\;j>k$ (26)
$\displaystyle[\mathsf{r}_{1j},\,\mathsf{z}]$
$\displaystyle=[\mathsf{r}_{2j}^{-1},\,\mathsf{z}]$

  * •

Conjugacy action of $\mathsf{t}_{1j}$

(27) $\displaystyle[\mathsf{t}_{1j},\,\mathsf{r}_{2k}]$ $\displaystyle=1$
$\displaystyle\mathrm{if}\;\;j<k$ (28)
$\displaystyle[\mathsf{t}_{1j},\,\mathsf{r}_{2j}]$
$\displaystyle=\mathsf{t}_{2j}^{-1}\,\mathsf{z}\,\mathsf{t}_{2j}$ (29)
$\displaystyle[\mathsf{t}_{1j},\,\mathsf{r}_{2k}]$
$\displaystyle=[\mathsf{t}_{2j}^{-1},\,\mathsf{z}]\;\;$
$\displaystyle\mathrm{if}\;\;j>k$ (31)
$\displaystyle[\mathsf{t}_{1j},\,\mathsf{t}_{2k}]$ $\displaystyle=1$
$\displaystyle\mathrm{if}\;\;j<k$ (32)
$\displaystyle[\mathsf{t}_{1j},\,\mathsf{t}_{2j}]$
$\displaystyle=[\mathsf{t}_{2j}^{-1},\,\mathsf{z}]$ (33)
$\displaystyle[\mathsf{t}_{1j},\,\mathsf{t}_{2k}]$
$\displaystyle=\mathsf{t}_{2j}^{-1}\,\mathsf{z}\,\mathsf{t}_{2j}\,\mathsf{z}^{-1}\,\mathsf{t}_{2k}\,\mathsf{z}\,\mathsf{t}_{2j}^{-1}\,\mathsf{z}^{-1}\,\mathsf{t}_{2j}\,\mathsf{t}_{2k}^{-1}\;\;$
$\displaystyle\mathrm{if}\;\;j>k$ (35)
$\displaystyle[\mathsf{t}_{1j},\,\mathsf{z}]$
$\displaystyle=[\mathsf{t}_{2j}^{-1},\,\mathsf{z}]$

######  Remark 2.2.

From (18) and (27) we can infer the corresponding conjugacy actions of
$\mathsf{r}_{1j}^{-1}$ and $\mathsf{t}_{1j}^{-1}$. We leave the cumbersome but
standard computations to the reader.

######  Remark 2.3.

Abelian groups admit no diagonal double Kodaira structures. Indeed, the
relation $[\mathsf{r}_{1j},\,\mathsf{t}_{2j}]=\mathsf{z}^{-1}$ in (18)
provides a non-trivial commutator in $G$, because $o(\mathsf{z})=n$.

######  Remark 2.4.

If $[G,\,G]\subseteq Z(G)$, then the relations defining a diagonal double
Kodaira structure of type $(b,\,n)$ assume the following simplified form.

  * •

Relations expressing the centrality of $\mathsf{z}$

(36)
$[\mathsf{r}_{1j},\mathsf{z}]=[\mathsf{t}_{1j},\mathsf{z}]=[\mathsf{r}_{2j},\mathsf{z}]=[\mathsf{t}_{2j},\mathsf{z}]=1$

  * •

Surface relations

(37)
$\displaystyle[\mathsf{r}_{1b}^{-1},\,\mathsf{t}_{1b}^{-1}]\,[\mathsf{r}_{1\,b-1}^{-1},\,\mathsf{t}_{1\,b-1}^{-1}]\,\cdots[\mathsf{r}_{11}^{-1},\,\mathsf{t}_{11}^{-1}]\,=\mathsf{z}$
(38)
$\displaystyle[\mathsf{r}_{21}^{-1},\,\mathsf{t}_{21}]\,[\mathsf{r}_{22}^{-1},\,\mathsf{t}_{22}]\cdots[\mathsf{r}_{2b}^{-1},\,\mathsf{t}_{2b}]=\mathsf{z}^{-1}$

  * •

Conjugacy action of $\mathsf{r}_{1j}$

(39) $\displaystyle[\mathsf{r}_{1j},\,\mathsf{r}_{2k}]$ $\displaystyle=1$
$\displaystyle\mathrm{for\;all}\;\;j,\,k$ (40)
$\displaystyle[\mathsf{r}_{1j},\,\mathsf{t}_{2k}]$
$\displaystyle=\mathsf{z}^{-\delta_{jk}}$

  * •

Conjugacy action of $\mathsf{t}_{1j}$

(42) $\displaystyle[\mathsf{t}_{1j},\,\mathsf{r}_{2k}]$
$\displaystyle=\mathsf{z}^{\delta_{jk}}$ (43)
$\displaystyle[\mathsf{t}_{1j},\,\mathsf{t}_{2k}]$ $\displaystyle=1$
$\displaystyle\mathrm{for\;all}\;\;j,\,k$

where $\delta_{jk}$ stands for the Kronecker symbol. Note that, being $G$ non-
abelian by Remark 2.3, the condition $[G,\,G]\subseteq Z(G)$ is equivalent to
$G$ having nilpotency class $2$, see [Is08, p. 22].

The definition of diagonal double Kodaira structure can be motivated by means
of some well-known concepts in geometric topology. Let $\Sigma_{b}$ be a
closed Riemann surface of genus $b$ and let $\mathscr{P}=(p_{1},\,p_{2})$ be
an ordered set of two distinct points on it. Let
$\Delta\subset\Sigma_{b}\times\Sigma_{b}$ be the diagonal. We denote by
$\mathsf{P}_{2}(\Sigma_{b})$ the _pure braid group_ of genus $b$ on two
strands, which is isomorphic to the fundamental group
$\pi_{1}(\Sigma_{b}\times\Sigma_{b}-\Delta,\,\mathscr{P})$. By Gonçalves-
Guaschi’s presentation of surface pure braid groups, see [GG04, Theorem 7],
[CaPol19, Theorem 1.7], we see that $\mathsf{P}_{2}(\Sigma_{b})$ can be
generated by $4g+1$ elements

(45) $\rho_{11},\,\tau_{11},\ldots,\rho_{1b},\,\tau_{1b},\,A_{12}$

subject to the following set of relations.

  * •

Surface relations

(46)
$\displaystyle[\rho_{1b}^{-1},\,\tau_{1b}^{-1}]\,\tau_{1b}^{-1}\,[\rho_{1\,b-1}^{-1},\,\tau_{1\,b-1}^{-1}]\,\tau_{1\,b-1}^{-1}\cdots[\rho_{11}^{-1},\,\tau_{11}^{-1}]\,\tau_{11}^{-1}\,(\tau_{11}\,\tau_{12}\cdots\tau_{1b})=A_{12}$
(47)
$\displaystyle[\rho_{21}^{-1},\,\tau_{21}]\,\tau_{21}\,[\rho_{22}^{-1},\,\tau_{22}]\,\tau_{22}\cdots[\rho_{2b}^{-1},\,\tau_{2b}]\,\tau_{2b}\,(\tau_{2b}^{-1}\,\tau_{2\,b-1}^{-1}\cdots\tau_{21}^{-1})=A_{12}^{-1}$

  * •

Conjugacy action of $\rho_{1j}$

(48) $\displaystyle[\rho_{1j},\,\rho_{2k}]$ $\displaystyle=1$
$\displaystyle\mathrm{if}\;\;j<k$ (49) $\displaystyle[\rho_{1j},\,\rho_{2j}]$
$\displaystyle=1$ (50) $\displaystyle[\rho_{1j},\,\rho_{2k}]$
$\displaystyle=A_{12}^{-1}\,\rho_{2k}\,\rho_{2j}^{-1}\,A_{12}\,\rho_{2j}\,\rho_{2k}^{-1}\;\;$
$\displaystyle\mathrm{if}\;\;j>k$ (52) $\displaystyle[\rho_{1j},\,\tau_{2k}]$
$\displaystyle=1$ $\displaystyle\mathrm{if}\;\;j<k$ (53)
$\displaystyle[\rho_{1j},\,\tau_{2j}]$ $\displaystyle=A_{12}^{-1}$ (54)
$\displaystyle[\rho_{1j},\,\tau_{2k}]$
$\displaystyle=[A_{12}^{-1},\,\tau_{2k}]$ $\displaystyle\mathrm{if}\;\;j>k$
(56) $\displaystyle[\rho_{1j},\,A_{12}]$
$\displaystyle=[\rho_{2j}^{-1},\,A_{12}]$

  * •

Conjugacy action of $\tau_{1j}$

(57) $\displaystyle[\tau_{1j},\,\rho_{2k}]$ $\displaystyle=1$
$\displaystyle\mathrm{if}\;\;j<k$ (58) $\displaystyle[\tau_{1j},\,\rho_{2j}]$
$\displaystyle=\tau_{2j}^{-1}\,A_{12}\,\tau_{2j}$ (59)
$\displaystyle[\tau_{1j},\,\rho_{2k}]$
$\displaystyle=[\tau_{2j}^{-1},\,A_{12}]\;\;$
$\displaystyle\mathrm{if}\;\;j>k$ (61) $\displaystyle[\tau_{1j},\,\tau_{2k}]$
$\displaystyle=1$ $\displaystyle\mathrm{if}\;\;j<k$ (62)
$\displaystyle[\tau_{1j},\,\tau_{2j}]$
$\displaystyle=[\tau_{2j}^{-1},\,A_{12}]$ (63)
$\displaystyle[\tau_{1j},\,\tau_{2k}]$
$\displaystyle=\tau_{2j}^{-1}\,A_{12}\,\tau_{2j}\,A_{12}^{-1}\,\tau_{2k}\,A_{12}\,\tau_{2j}^{-1}\,A_{12}^{-1}\,\tau_{2j}\,\tau_{2k}^{-1}\;\;$
$\displaystyle\mathrm{if}\;\;j>k$ (65) $\displaystyle[\tau_{1j},\,A_{12}]$
$\displaystyle=[\tau_{2j}^{-1},\,A_{12}]$

Here the elements $\rho_{ij}$ and $\tau_{ij}$ are the braids depicted in
Figure 1, whereas $A_{12}$ is the braid depicted in Figure 2.

Figure 1. The pure braids $\rho_{1j}$ and $\rho_{2j}$ on $\Sigma_{b}$. If
$\ell\neq i$, the path corresponding to $\rho_{ij}$ and $\tau_{ij}$ based at
$p_{\ell}$ is the constant path. Figure 2. The pure braid $A_{12}$ on
$\Sigma_{b}$

######  Remark 2.5.

Under the identification of $\mathsf{P}_{2}(\Sigma_{b})$ with
$\pi_{1}(\Sigma_{b}\times\Sigma_{b}-\Delta,\,\mathscr{P})$, the generator
$A_{12}\in\mathsf{P}(\Sigma_{b})$ represents the homotopy class
$\gamma_{\Delta}\in\pi_{1}(\Sigma_{b}\times\Sigma_{b}-\Delta,\,\mathscr{P})$
of a loop in $\Sigma_{b}\times\Sigma_{b}$ that “winds once” around the
diagonal $\Delta$.

We can now state the following

######  Proposition 2.6.

A finite group $G$ admits a diagonal double Kodaira structure of type
$(b,\,n)$ if and only if there is a surjective group homomorphism

(66) $\varphi\colon\mathsf{P}_{2}(\Sigma_{b})\longrightarrow G$

such that $\varphi(A_{12})$ has order $n$.

###### Proof.

If such a $\varphi\colon\mathsf{P}_{2}(\Sigma_{b})\longrightarrow G$ exists,
we can obtain a diagonal double Kodaira structure on $G$ by setting

(67)
$\mathsf{r}_{ij}=\varphi(\rho_{ij}),\quad\mathsf{t}_{ij}=\varphi(\tau_{ij}),\quad\mathsf{z}=\varphi({A_{12}}).$

Conversely, if $G$ admits a diagonal double Kodaira structure, then (67)
defines a group homomorphism
$\varphi\colon\mathsf{P}_{2}(\Sigma_{b})\longrightarrow G$ with the desired
properties. ∎

The braid group $\mathsf{P}_{2}(\Sigma_{b})$ is the middle term of two split
short exact sequences

(68)
$1\longrightarrow\pi_{1}(\Sigma_{b}-\\{p_{i}\\},\,p_{j})\longrightarrow\mathsf{P}_{2}(\Sigma_{b})\longrightarrow\pi_{1}(\Sigma_{b},\,p_{i})\longrightarrow
1,$

where $\\{i,\,j\\}=\\{1,\,2\\}$, induced by the two natural projections of
pointed topological spaces

(69)
$(\Sigma_{b}\times\Sigma_{b}-\Delta,\,\mathscr{P})\longrightarrow(\Sigma_{b},\,p_{i}),$

see [GG04, Theorem 1]. Since we have

(70)
$\begin{split}\pi_{1}(\Sigma_{b}-\\{p_{2}\\},\,p_{1})&=\langle\rho_{11},\,\tau_{11},\ldots,\rho_{1b},\,\tau_{1b},\;A_{12}\rangle\\\
\pi_{1}(\Sigma_{b}-\\{p_{1}\\},\,p_{2})&=\langle\rho_{21},\,\tau_{21},\ldots,\rho_{2b},\,\tau_{2b},\;A_{12}\rangle,\end{split}$

it follows that the two subgroups

(71)
$\begin{split}K_{1}&:=\langle\mathsf{r}_{11},\,\mathsf{t}_{11},\ldots,\mathsf{r}_{1b},\,\mathsf{t}_{1b},\;\mathsf{z}\rangle\\\
K_{2}&:=\langle\mathsf{r}_{21},\,\mathsf{t}_{21},\ldots,\mathsf{r}_{2b},\,\mathsf{t}_{2b},\;\mathsf{z}\rangle\end{split}$

are both normal in $G$, and that there are two short exact sequences

(72) $\begin{split}&1\longrightarrow K_{1}\longrightarrow G\longrightarrow
Q_{2}\longrightarrow 1\\\ &1\longrightarrow K_{2}\longrightarrow
G\longrightarrow Q_{1}\longrightarrow 1,\end{split}$

such the elements
$\mathsf{r}_{21},\,\mathsf{t}_{21},\ldots,\mathsf{r}_{2b},\,\mathsf{t}_{2b}$
yield a complete system of coset representatives for $Q_{2}$, whereas the
elements
$\mathsf{r}_{11},\,\mathsf{t}_{11},\ldots,\mathsf{r}_{1b},\,\mathsf{t}_{1b}$
yield a complete system of coset representatives for $Q_{1}$.

Let us now give a couple of definitions, whose geometrical meaning will become
clear in Section 4, see in particular Proposition 4.3 and Remark 4.4.

######  Definition 2.7.

Let $\mathfrak{S}$ be a diagonal double Kodaira structure of type $(b,\,n)$ on
a finite group $G$. Its _signature_ is defined as

(73)
$\sigma(\mathfrak{S})=\frac{1}{3}\,|G|\,(2b-2)\left(1-\frac{1}{n^{2}}\right).$

######  Definition 2.8.

A diagonal double Kodaira structure on $G$ is called _strong_ if
$K_{1}=K_{2}=G$.

For later use, let us write down the special case consisting of a diagonal
double Kodaira structure of type $(2,\,n)$. It is an ordered set of nine
generators of $G$

(74)
$(\mathsf{r}_{11},\,\mathsf{t}_{11},\,\mathsf{r}_{12},\,\mathsf{t}_{12},\,\mathsf{r}_{21},\,\mathsf{t}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22},\,\mathsf{z}),$

with $o(\mathsf{z})=n$, subject to the following relations.

(75) $\displaystyle\mathbf{(S1)}$
$\displaystyle\,\,[\mathsf{r}_{12}^{-1},\,\mathsf{t}_{12}^{-1}]\,\mathsf{t}_{12}^{-1}\,[\mathsf{r}_{11}^{-1},\,\mathsf{t}_{11}^{-1}]\,\mathsf{t}_{11}^{-1}\,(\mathsf{t}_{11}\,\mathsf{t}_{12})=\mathsf{z}$
$\displaystyle\mathbf{(S2)}$
$\displaystyle\,\,[\mathsf{r}_{21}^{-1},\,\mathsf{t}_{21}]\;\mathsf{t}_{21}\;[\mathsf{r}_{22}^{-1},\,\mathsf{t}_{22}]\,\mathsf{t}_{22}\,(\mathsf{t}_{22}^{-1}\,\mathsf{t}_{21}^{-1})=\mathsf{z}^{-1}$
$\displaystyle\mathbf{(R1)}$
$\displaystyle\,\,[\mathsf{r}_{11},\,\mathsf{r}_{22}]=1$
$\displaystyle\mathbf{(R6)}$
$\displaystyle\,\,[\mathsf{r}_{12},\,\mathsf{r}_{22}]=1$
$\displaystyle\mathbf{(R2)}$
$\displaystyle\,\,[\mathsf{r}_{11},\,\mathsf{r}_{21}]=1$
$\displaystyle\mathbf{(R7)}$
$\displaystyle\,\,[\mathsf{r}_{12},\,\mathsf{r}_{21}]=\mathsf{z}^{-1}\,\mathsf{r}_{21}\,\mathsf{r}_{22}^{-1}\,\mathsf{z}\,\mathsf{r}_{22}\,\mathsf{r}_{21}^{-1}$
$\displaystyle\mathbf{(R3)}$
$\displaystyle\,\,[\mathsf{r}_{11},\,\mathsf{t}_{22}]=1$
$\displaystyle\mathbf{(R8)}$
$\displaystyle\,\,[\mathsf{r}_{12},\,\mathsf{t}_{22}]=\mathsf{z}^{-1}$
$\displaystyle\mathbf{(R4)}$
$\displaystyle\,\,[\mathsf{r}_{11},\,\mathsf{t}_{21}]=\mathsf{z}^{-1}$
$\displaystyle\mathbf{(R9)}$
$\displaystyle\,\,[\mathsf{r}_{12},\,\mathsf{t}_{21}]=[\mathsf{z}^{-1},\,\mathsf{t}_{21}]$
$\displaystyle\mathbf{(R5)}$
$\displaystyle\,\,[\mathsf{r}_{11},\,\mathsf{z}]=[\mathsf{r}_{21}^{-1},\,\mathsf{z}]$
$\displaystyle\mathbf{(R10)}$
$\displaystyle\,\,[\mathsf{r}_{12},\,\mathsf{z}]=[\mathsf{r}_{22}^{-1},\,\mathsf{z}]$
$\displaystyle\mathbf{(T1)}$
$\displaystyle\,\,[\mathsf{t}_{11},\,\mathsf{r}_{22}]=1$
$\displaystyle\mathbf{(T6)}$
$\displaystyle\,\,[\mathsf{t}_{12},\,\mathsf{r}_{22}]=\mathsf{t}_{22}^{-1}\,\mathsf{z}\,\mathsf{t}_{22}$
$\displaystyle\mathbf{(T2)}$
$\displaystyle\,\,[\mathsf{t}_{11},\,\mathsf{r}_{21}]=\mathsf{t}_{21}^{-1}\,\mathsf{z}\,\mathsf{t}_{21}$
$\displaystyle\mathbf{(T7)}$
$\displaystyle\,\,[\mathsf{t}_{12},\,\mathsf{r}_{21}]=[\mathsf{t}_{22}^{-1},\,\mathsf{z}]$
$\displaystyle\mathbf{(T3)}$
$\displaystyle\,\,[\mathsf{t}_{11},\,\mathsf{t}_{22}]=1$
$\displaystyle\mathbf{(T8)}$
$\displaystyle\,\,[\mathsf{t}_{12},\,\mathsf{t}_{22}]=[\mathsf{t}_{22}^{-1},\,\mathsf{z}]$
$\displaystyle\mathbf{(T4)}$
$\displaystyle\,\,[\mathsf{t}_{11},\,\mathsf{t}_{21}]=[\mathsf{t}_{21}^{-1},\,\mathsf{z}]$
$\displaystyle\mathbf{(T9)}$
$\displaystyle\,\,[\mathsf{t}_{12},\,\mathsf{t}_{21}]=\mathsf{t}_{22}^{-1}\,\mathsf{z}\,\mathsf{t}_{22}\,\mathsf{z}^{-1}\,\mathsf{t}_{21}\,\mathsf{z}\,\mathsf{t}_{22}^{-1}\,\mathsf{z}^{-1}\,\mathsf{t}_{22}\,\mathsf{t}_{21}^{-1}$
$\displaystyle\mathbf{(T5)}$
$\displaystyle\,\,[\mathsf{t}_{11},\,\mathsf{z}]=[\mathsf{t}_{21}^{-1},\,\mathsf{z}]$
$\displaystyle\mathbf{(T10)}$
$\displaystyle\,\,[\mathsf{t}_{12},\,\mathsf{z}]=[\mathsf{t}_{22}^{-1},\,\mathsf{z}]$

######  Remark 2.9.

When $[G,\,G]\subseteq Z(G)$, we have

(76)
$\begin{split}&[\mathsf{r}_{11},\,\mathsf{z}]=[\mathsf{t}_{11},\,\mathsf{z}]=[\mathsf{r}_{12},\,\mathsf{z}]=[\mathsf{t}_{12},\,\mathsf{z}]=1\\\
&[\mathsf{r}_{21},\,\mathsf{z}]=[\mathsf{t}_{21},\,\mathsf{z}]=[\mathsf{r}_{22},\,\mathsf{z}]=[\mathsf{t}_{22},\,\mathsf{z}]=1\end{split}$

and the previous relations become

(77) $\displaystyle\mathbf{(S1^{\prime})}$
$\displaystyle\,\,[\mathsf{r}_{12}^{-1},\,\mathsf{t}_{12}^{-1}]\,[\mathsf{r}_{11}^{-1},\,\mathsf{t}_{11}^{-1}]=\mathsf{z}$
$\displaystyle\mathbf{(S2^{\prime})}$
$\displaystyle\,\,[\mathsf{r}_{21}^{-1},\,\mathsf{t}_{21}]\;[\mathsf{r}_{22}^{-1},\,\mathsf{t}_{22}]=\mathsf{z}^{-1}$
$\displaystyle\mathbf{(R1^{\prime})}$
$\displaystyle\,\,[\mathsf{r}_{11},\,\mathsf{r}_{22}]=1$
$\displaystyle\mathbf{(R6^{\prime})}$
$\displaystyle\,\,[\mathsf{r}_{12},\,\mathsf{r}_{22}]=1$
$\displaystyle\mathbf{(R2^{\prime})}$
$\displaystyle\,\,[\mathsf{r}_{11},\,\mathsf{r}_{21}]=1$
$\displaystyle\mathbf{(R7^{\prime})}$
$\displaystyle\,\,[\mathsf{r}_{12},\,\mathsf{r}_{21}]=1$
$\displaystyle\mathbf{(R3^{\prime})}$
$\displaystyle\,\,[\mathsf{r}_{11},\,\mathsf{t}_{22}]=1$
$\displaystyle\mathbf{(R8^{\prime})}$
$\displaystyle\,\,[\mathsf{r}_{12},\,\mathsf{t}_{22}]=\mathsf{z}^{-1}$
$\displaystyle\mathbf{(R4^{\prime})}$
$\displaystyle\,\,[\mathsf{r}_{11},\,\mathsf{t}_{21}]=\mathsf{z}^{-1}$
$\displaystyle\mathbf{(R9^{\prime})}$
$\displaystyle\,\,[\mathsf{r}_{12},\,\mathsf{t}_{21}]=1$
$\displaystyle\mathbf{(T1^{\prime})}$
$\displaystyle\,\,[\mathsf{t}_{11},\,\mathsf{r}_{22}]=1$
$\displaystyle\mathbf{(T6^{\prime})}$
$\displaystyle\,\,[\mathsf{t}_{12},\,\mathsf{r}_{22}]=\mathsf{z}$
$\displaystyle\mathbf{(T2^{\prime})}$
$\displaystyle\,\,[\mathsf{t}_{11},\,\mathsf{r}_{21}]=\mathsf{z}$
$\displaystyle\mathbf{(T7^{\prime})}$
$\displaystyle\,\,[\mathsf{t}_{12},\,\mathsf{r}_{21}]=1$
$\displaystyle\mathbf{(T3^{\prime})}$
$\displaystyle\,\,[\mathsf{t}_{11},\,\mathsf{t}_{22}]=1$
$\displaystyle\mathbf{(T8^{\prime})}$
$\displaystyle\,\,[\mathsf{t}_{12},\,\mathsf{t}_{22}]=1$
$\displaystyle\mathbf{(T4^{\prime})}$
$\displaystyle\,\,[\mathsf{t}_{11},\,\mathsf{t}_{21}]=1$
$\displaystyle\mathbf{(T9^{\prime})}$
$\displaystyle\,\,[\mathsf{t}_{12},\,\mathsf{t}_{21}]=1$

##  3\. Structures on groups of order at most $32$

###  3.1. Prestructures

######  Definition 3.1.

Let $G$ be a finite group. A _prestructure_ on $G$ is an ordered set of nine
elements

(78)
$(\mathsf{r}_{11},\,\mathsf{t}_{11},\,\mathsf{r}_{12},\,\mathsf{t}_{12},\,\mathsf{r}_{21},\,\mathsf{t}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22},\,\mathsf{z}),$

with $o(\mathsf{z})=n\geq 2$, subject to the relations
$(\mathrm{R1}),\ldots,(\mathrm{R10})$, $(\mathrm{T1}),\ldots,(\mathrm{T10})$
in (75).

In other words, the nine elements must satisfy all the relations defining a
diagonal double Kodaira structure of type $(2,\,n)$, except the surface
relations. In particular, no abelian group admits prestructures. Note that we
are _not_ requiring that the elements of the prestructure generate $G$.

######  Proposition 3.2.

If a finite group $G$ admits a diagonal double Kodaira structure of type
$(b,\,n)$, then it admits a prestructure with $o(\mathsf{z})=n$.

###### Proof.

Consider the ordered set of nine elements
$(\mathsf{r}_{11},\,\mathsf{t}_{11},\,\mathsf{r}_{12},\,\mathsf{t}_{12},\,\mathsf{r}_{21},\,\mathsf{t}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22},\,\mathsf{z})$
in Definition (2.1) and the relations satisfied by them, with the exception of
the surface relations. ∎

######  Remark 3.3.

Let $G$ be a finite group that admits a prestructure. Then $\mathsf{z}$ and
all its conjugates are non-trivial elements of $G$ and so, from relations
$\mathrm{(R4)}$, $\mathrm{(R8)}$, $\mathrm{(T2)}$, $\mathrm{(T6)}$, it follows
that $\mathsf{r}_{11},\,\mathsf{r}_{12},\,\mathsf{r}_{21},\,\mathsf{r}_{22}$
and $\mathsf{t}_{12},\,\mathsf{t}_{12},\,\mathsf{t}_{21},\mathsf{t}_{22}$ are
non-central elements of $G$.

######  Proposition 3.4.

If $G$ is a _CCT_ -group, then $G$ admits no prestructures and, subsequently,
no diagonal double Kodaira structures.

###### Proof.

The second statement is a direct consequence of the first one (Proposition
3.2), hence it suffices then to check that $G$ admits no prestructures.
Otherwise, keeping in mind Remark 3.3, we see that $\mathrm{(R6)}$ and
$\mathrm{(T1)}$ imply $[\mathsf{r}_{12},\,\mathsf{t}_{11}]=1$. From this and
$\mathrm{(T3)}$ we get $[\mathsf{r}_{12},\,\mathsf{t}_{22}]=1$, that
contradicts $\mathrm{(R8)}$. ∎

Given a finite group $G$, we define the _socle_ of $G$, denoted by
$\mathrm{soc}(G)$, as the intersection of all non-trivial, normal subgroups of
$G$. For instance, $G$ is simple if and only if $\mathrm{soc}(G)=G$.

######  Definition 3.5.

A finite group $G$ is called _monolithic_ if $\mathrm{soc}(G)\neq\\{1\\}$.
Equivalently, $G$ is monolithic if it contains precisely one minimal non-
trivial, normal subgroup.

######  Example 3.6.

If $G$ is an extra-special $p$-group, then $G$ is monolithic and
$\operatorname{soc}(G)=Z(G)$. Indeed, since $Z(G)\simeq\mathbb{Z}_{p}$ is
normal in $G$, by definition of socle we always have
$\operatorname{soc}(G)\subseteq Z(G)$. On the other hand, every non-trivial,
normal subgroup of an extra-special group contains the center (see [Rob96,
Exercise 9 p. 146]), hence $Z(G)\subseteq\operatorname{soc}(G)$.

######  Proposition 3.7.

The following holds.

  * $\boldsymbol{(1)}$

Assume that $G$ admits a prestructure, whereas no proper quotient of $G$ does.
Then $G$ is monolithic and $\mathsf{z}\in\mathrm{soc}(G)$.

  * $\boldsymbol{(2)}$

Assume that $G$ admits a prestructure, whereas no proper subgroup of $G$ does.
Then the elements of the prestructure generate $G$.

###### Proof.

$\boldsymbol{(1)}$ Let
$\mathfrak{S}=(\mathsf{r}_{11},\,\mathsf{t}_{11},\,\mathsf{r}_{12},\,\mathsf{t}_{12},\,\mathsf{r}_{21},\,\mathsf{t}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22},\,\mathsf{z})$
be a prestructure in $G$. Assume that there is a non-trivial normal subgroup
$N$ of $G$ such that $\mathsf{z}\notin N$. Then $\mathsf{\bar{z}}\in G/N$ is
non-trivial, and so
$\bar{\mathfrak{S}}=(\mathsf{\bar{r}}_{11},\,\mathsf{\bar{t}}_{11},\,\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12},\,\mathsf{\bar{r}}_{21},\,\mathsf{\bar{t}}_{21},\,\mathsf{\bar{r}}_{22},\,\mathsf{\bar{t}}_{22},\,\mathsf{\bar{z}})$
is a prestructure in the quotient group $G/N$, contradiction. Therefore we
must have $\mathsf{z}\in\mathrm{soc}(G)$, in particular, $G$ is monolithic.

$\boldsymbol{(2)}$ Clear, because every prestructure $\mathfrak{S}$ in $G$ is
also a prestructure in the subgroup $\langle\mathfrak{S}\rangle$. ∎

######  Corollary 3.8.

Given a prestructure on an extra-special $p$-group $G$, the element
$\mathsf{z}$ is a generator of $Z(G)\simeq\mathbb{Z}_{p}$.

###### Proof.

If $G$ is extra-special, every proper quotient of $G$ is abelian, hence it
admits no prestructures. The result now follows from Example 3.6 and
Proposition 3.7 (1). ∎

Note that, by Corollary 3.8, in the case of extra-special $p$-groups the
choice of calling $\mathsf{z}$ the element in the prestructure is coherent
with presentations (12) and (13). The case of diagonal double Kodaira
structures on extra-special groups of order $32$ will be studied in Subsection
3.4.

###  3.2. The case $|G|<32$

######  Proposition 3.9.

If $|G|<32$, then $G$ admits no diagonal double Kodaira structures.

###### Proof.

By Corollary 1.6, Proposition 1.7 and Proposition 3.4, it remains only to
check that the symmetric group $\mathsf{S}_{4}$ admits no prestructures. We
start by observing that

$\mathrm{soc}(\mathsf{S}_{4})=\mathsf{V}_{4}=\langle(1\,2)(3\,4),\,(1\,3)(2\,4)\rangle$

and so, by part $\mathrm{(1)}$ of Proposition 3.7, if $\mathfrak{S}$ is a
prestructure on $\mathsf{S}_{4}$ then $\mathsf{z}\in\mathsf{V}_{4}$. Let
$\mathsf{x},\,\mathsf{y}\in\mathsf{S}_{4}$ be such that
$[\mathsf{x},\,\mathsf{y}]=\mathsf{z}$. Examining the tables of subgroups of
$\mathsf{S}_{4}$ given in [S4], by straightforward computations and keeping in
mind that the cycle type determines the conjugacy class, we deduce that either
$\mathsf{x},\,\mathsf{y}\in
C_{\mathsf{S}_{4}}(\mathsf{z})\simeq\mathsf{D}_{8}$ or
$\mathsf{x},\,\mathsf{y}\in\mathsf{A}_{4}$. Every pair in $\mathsf{A}_{4}$
includes at least a $3$-cycle and so, if
$[\mathsf{x},\,\mathsf{y}]=\mathsf{z}$ and both $\mathsf{x}$ and $\mathsf{y}$
have even order, then $\mathsf{x}$ and $\mathsf{y}$ centralize $\mathsf{z}$.

If $\mathsf{x}\in\mathsf{S}_{4}$ is a $3$-cycle, then
$C_{\mathsf{S}_{4}}(\mathsf{x})=\langle\mathsf{x}\rangle\simeq\mathbb{Z}_{3}$.
So, from relations $(\mathrm{R1})$, $(\mathrm{R2})$, $(\mathrm{R3})$,
$(\mathrm{R6})$, it follows that, if one of the elements
$\mathsf{r}_{11},\,\mathsf{r}_{12},\,\mathsf{r}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22}$
is a $3$-cycle, then all these elements generate the same cyclic subgroup.
This contradicts $(\mathrm{R8})$, hence
$\mathsf{r}_{11},\,\mathsf{r}_{12},\,\mathsf{r}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22}$
all have even order.

Let us look now at relation $(\mathrm{R8})$. Since
$\mathsf{r}_{12},\,\mathsf{t}_{22}$ have even order, from the previous remark
we infer $\mathsf{r}_{12},\,\mathsf{t}_{22}\in
C_{\mathsf{S}_{4}}(\mathsf{z})$. Let us consider $\mathsf{r}_{11}$. If
$\mathsf{r}_{11}$ belongs to $\mathsf{A}_{4}$, being an element of even order
it must be conjugate to $\mathsf{z}$, and so it commutes with $\mathsf{z}$;
otherwise, by $(\mathrm{R4})$, both $\mathsf{r}_{11}$ and $\mathsf{t}_{21}$
commute with $\mathsf{z}$. Summing up, in any case we have $\mathsf{r}_{11}\in
C_{\mathsf{S}_{4}}(\mathsf{z})$.

Relation $(\mathrm{R5})$ can be rewritten as
$\mathsf{r}_{11}\mathsf{r}_{21}\in C_{\mathsf{S}_{4}}(\mathsf{z})$, hence
$\mathsf{r}_{21}\in C_{\mathsf{S}_{4}}(\mathsf{z})$. Analogously, relation
$(\mathrm{R10})$ can be rewritten as $\mathsf{r}_{12}\mathsf{r}_{22}\in
C_{\mathsf{S}_{4}}(\mathsf{z})$, hence $\mathsf{r}_{22}\in
C_{\mathsf{S}_{4}}(\mathsf{z})$.

Using relation $\mathrm{(R9)}$, we get $\mathsf{r}_{12}\mathsf{z}\in
C_{\mathsf{S}_{4}}(\mathsf{t}_{21})$. Since $\mathsf{r}_{12}$ and $\mathsf{z}$
commute and their orders are powers of $2$, it follows that
$o(\mathsf{r}_{12}\mathsf{z})$ is also a power of $2$. Therefore
$\mathsf{t}_{21}$ cannot be a $3$-cycle, otherwise
$C_{\mathsf{S}_{4}}(\mathsf{t}_{21})\simeq\mathbb{Z}_{3}$ and so
$\mathsf{r}_{12}\mathsf{z}=1$ that, in turn, would imply
$[\mathsf{r}_{12},\,\mathsf{t}_{22}]=1$, contradicting $(\mathrm{R8})$. It
follows that $\mathsf{t}_{21}$ has even order and so, since $\mathsf{r}_{11}$
has even order as well, by $\mathrm{(R4)}$ we infer $\mathsf{t}_{21}\in
C_{\mathsf{S}_{4}}(z)$.

Now we can rewrite $(\mathrm{T2})$ as
$[\mathsf{t}_{11},\,\mathsf{r}_{21}]=\mathsf{z}$. If $\mathsf{t}_{11}$ were a
$3$-cycle, from $(\mathrm{T1})$ we would get $\mathsf{r}_{22}\in
C_{\mathsf{S}_{4}}(\mathsf{t}_{11})\simeq\mathbb{Z}_{3}$, a contradiction
since $\mathsf{r}_{22}$ has even order. Thus $\mathsf{t}_{11}$ has even order
and so it belongs to $C_{\mathsf{S}_{4}}(z)$, because $\mathsf{r}_{21}$ has
even order, too. Analogously, by using $(\mathrm{T6})$ and $(\mathrm{T7})$, we
infer $\mathsf{t}_{12}\in C_{\mathsf{S}_{4}}(z)$.

Summarizing, if $\mathfrak{S}$ were a prestructure on $\mathsf{S}_{4}$ we
should have

(79) $\langle\mathfrak{S}\rangle=C_{\mathsf{S}_{4}}(z)\simeq\mathsf{D}_{8},$

contradicting part $(\mathrm{2})$ of Proposition 3.7. ∎

###  3.3. The case $|G|=32$ and $G$ non-extra-special

We start by proving the following partial strengthening of Proposition 3.4.

######  Proposition 3.10.

Let $G$ be a non-abelian finite group, and let $H$ be the subgroup of $G$
generated by those elements whose centralizer is non-abelian. If $H$ is
abelian and $[H:Z(G)]\leq 4$, then $G$ admits no prestructures with
$\mathsf{z}\in Z(G)$.

###### Proof.

First of all, remark that $Z(G)$ is a (normal) subgroup of $H$ because $G$ is
non-abelian. Assume now, by contradiction, that the elements
$(\mathsf{r}_{11},\,\mathsf{t}_{11},\,\mathsf{r}_{12},\,\mathsf{t}_{12},\,\mathsf{r}_{21},\,\mathsf{t}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22},\,\mathsf{z})$
form a prestructure on $G$, with $\mathsf{z}\in Z(G)$. Then these elements
satisfy relations $(\mathrm{R1}^{\prime}),\ldots,(\mathrm{R9}^{\prime})$,
$(\mathrm{T1}^{\prime}),\ldots,(\mathrm{T9}^{\prime})$ in (77). As $H$ is
abelian, $(\mathrm{R4}^{\prime})$ implies that at least one between
$\mathsf{r}_{11},\,\mathsf{t}_{21}$ does not belong to $H$.

Let us assume $\mathsf{r}_{11}\notin H$. Thus $C_{G}(\mathsf{r}_{11})$ is
abelian, and so $(\mathrm{R2}^{\prime})$ and $(\mathrm{R3}^{\prime})$ yield
$[\mathsf{r}_{21},\,\mathsf{t}_{22}]=1$. From this, using
$(\mathrm{T2}^{\prime})$ and $(\mathrm{T3}^{\prime})$, we infer that
$C_{G}(\mathsf{t}_{22})$ is non-abelian. Similar considerations show that
$C_{G}(\mathsf{r}_{21})$ and $C_{G}(\mathsf{r}_{22})$ are non-abelian, and so
we have $\mathsf{r}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22}\in H$. Using
$(\mathrm{T2}^{\prime})$, $(\mathrm{T6}^{\prime})$, $(\mathrm{R8}^{\prime})$,
together with the fact that $H$ is abelian, we deduce
$\mathsf{t}_{11},\,\mathsf{t}_{12},\,\mathsf{r}_{12}\notin H$. In particular,
$C_{G}(\mathsf{r}_{12})$ is abelian, so $(\mathrm{R7}^{\prime})$ and
$(\mathrm{R9}^{\prime})$ yield $[\mathsf{r}_{21},\,\mathsf{t}_{21}]=1$;
therefore $(\mathrm{T2}^{\prime})$ and $(\mathrm{T4}^{\prime})$ imply that
$C_{G}(\mathsf{t}_{21})$ is non-abelian, and so $\mathsf{t}_{21}\in H$.
Summing up, we have proved that the four elements
$\mathsf{r}_{21},\,\mathsf{t}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22}$ belong
to $H$; since they are all non-central, we infer that they yield four non-
trivial elements in the quotient group $H/Z(G)$. On the other hand, we have
$[H:Z(G)]\leq 4$, and so $H/Z(G)$ contains at most three non-trivial elements;
it follows that (at least) two among the elements
$\mathsf{r}_{21},\,\mathsf{t}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22}$ have
the same image in $H/Z(G)$. This means that these two elements are of the form
$g,\,gz$, with $z\in Z(G)$, and so they have the same centralizer. But this is
impossible: in fact, relations (77) show that each element in the set
$\\{\mathsf{r}_{21},\,\mathsf{t}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22}\\}$
fails to commute with exactly one element in the set
$\\{\mathsf{r}_{11},\,\mathsf{t}_{11},\,\mathsf{r}_{12},\,\mathsf{t}_{12}\\}$,
and no two elements in
$\\{\mathsf{r}_{21},\,\mathsf{t}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22}\\}$
fail to commute with the same element in
$\\{\mathsf{r}_{11},\,\mathsf{t}_{11},\,\mathsf{r}_{12},\,\mathsf{t}_{12}\\}$.

The remaining case, namely $\mathsf{t}_{21}\notin H$, can be dealt with in an
analogous way. Indeed, in this situation we obtain
$\\{\mathsf{r}_{11},\,\mathsf{t}_{11},\,\mathsf{r}_{12},\,\mathsf{t}_{12}\\}\subseteq
H$, that leads to a contradiction as before. ∎

We can now rule out the non-extra-special groups of order $32$.

######  Proposition 3.11.

Let $G$ be a finite group of order $32$ which is not extra-special. Then $G$
admits no diagonal double Kodaira structures.

###### Proof.

If $G$ is a CCT-group, then the result follows from Proposition 3.4. Thus, by
Proposition 1.14, we must only consider the cases $G=G(32,\,t)$, where
$t\in\\{6,\,7,\,8,\,43,\,44\\}$. Standard computations using the presentations
in Table 2 of Appendix A show that all these groups are monolithic, and that
for all of them $\mathrm{soc}(G)=Z(G)\simeq\mathbb{Z}_{2}$, cf. the proof of
Proposition 1.14. Since no proper quotients of $G$ admit diagonal double
Kodaira structures (Proposition 3.9), it follows from Proposition 3.7 that
every diagonal double Kodaira structure on $G$ is such that $\mathsf{z}$ is
the generator of $Z(G)$. Let $H$ be the subgroup of $G$ generated by those
elements whose centralizer is non-abelian; then, by Proposition 3.10, we are
done, provided that in every case $H$ is abelian and $[H\,:\,Z(G)]\leq 4$. Let
us now show that this is indeed true, leaving the straightforward computations
to the reader.

  * •

$G=G(32,\,6).$ In this case $\mathrm{soc}(G)=Z(G)=\langle x\rangle$ and
$H=\langle x,\,y,\,w^{2}\rangle$. Then $H\simeq(\mathbb{Z}_{2})^{3}$ and
$[H:Z(G)]=4$.

  * •

$G=G(32,\,7).$ In this case $\mathrm{soc}(G)=Z(G)=\langle w\rangle$ and
$H=\langle z,\,u,\,w\rangle$. Then $H\simeq\mathbb{Z}_{4}\times\mathbb{Z}_{2}$
and $[H:Z(G)]=4$.

  * •

$G=G(32,\,8).$ In this case $\mathrm{soc}(G)=Z(G)=\langle x^{4}\rangle$ and
$H=\langle x^{2},\,y,\,z^{2}\rangle$. Then
$H\simeq\mathbb{Z}_{4}\times\mathbb{Z}_{2}$ and $[H:Z(G)]=4$.

  * •

$G=G(32,\,43).$ In this case $\mathrm{soc}(G)=Z(G)=\langle x^{4}\rangle$ and
$H=\langle x^{2},\,z\rangle$. Then $H\simeq\mathbb{Z}_{4}\times\mathbb{Z}_{2}$
and $[H:Z(G)]=4$.

  * •

$G=G(32,\,44).$ In this case $\mathrm{soc}(G)=Z(G)=\langle i^{2}\rangle$ and
$H=\langle x,\,k\rangle$. Then $H\simeq\mathbb{Z}_{4}\times\mathbb{Z}_{2}$ and
$[H:Z(G)]=4$.

This completes the proof. ∎

###  3.4. The case $|G|=32$ and $G$ extra-special

We are now ready to address the case where $|G|=32$ and $G$ is extra-special.
Let us first recall some results on extra-special $p$-groups, referring the
reader to [Win72] for more details.

Let $G$ be an extra-special $p$-group of order $p^{2b+1}$ and
$\mathsf{x},\,\mathsf{y}\in G$. Setting
$(\bar{\mathsf{x}},\,\bar{\mathsf{y}})=\bar{a}$ where
$[\mathsf{x},\,\mathsf{y}]=\mathsf{z}^{a}$, the quotient group
$V=G/Z(G)\simeq(\mathbb{Z}_{p})^{2b}$ becomes a non-degenerate symplectic
vector space over $\mathbb{Z}_{p}$. Looking at $\eqref{eq:H5}$ and
$\eqref{eq:G5}$, we see that in both cases
$G=\mathsf{H}_{2b+1}(\mathbb{Z}_{p})$ and
$G=\mathsf{G}_{2b+1}(\mathbb{Z}_{p})$ we have

(80)
$(\bar{\mathsf{r}}_{j},\,\bar{\mathsf{r}}_{k})=0,\quad(\bar{\mathsf{t}}_{j},\,\bar{\mathsf{t}}_{k})=0,\quad(\bar{\mathsf{r}}_{j},\,\bar{\mathsf{t}}_{k})=-\delta_{jk}$

for all $j,\,k\in\\{1,\ldots,b\\}$, so that

(81)
$\bar{\mathsf{r}}_{1},\,\bar{\mathsf{t}}_{1},\ldots,\bar{\mathsf{r}}_{b},\,\bar{\mathsf{t}}_{b}$

is an ordered symplectic basis for $V\simeq(\mathbb{Z}_{p})^{2b}$. If $p=2$,
we can also set $q(\bar{\mathsf{x}})=\bar{c}$, where
$\mathsf{x}^{2}=\mathsf{z}^{c}$ and $c\in\\{0,\,1\\}$; this is a quadratic
form on $V$. If $\bar{\mathsf{x}}\in G/Z(G)$ is expressed in coordinates, with
respect to the symplectic basis (81), by the vector
$(\xi_{1},\,\psi_{1},\ldots,\xi_{b},\,\psi_{b})\in(\mathbb{Z}_{2})^{2b}$, then
a straightforward computation yields

(82)
$q(\bar{\mathsf{x}})=\begin{cases}\xi_{1}\psi_{1}+\cdots+\xi_{b}\psi_{b},&\textrm{if
}G=\mathsf{H}_{2b+1}(\mathbb{Z}_{2})\\\
\xi_{1}\psi_{1}+\cdots+\xi_{b}\psi_{b}+\xi_{b}^{2}+\psi_{b}^{2}&\textrm{if
}G=\mathsf{G}_{2b+1}(\mathbb{Z}_{2}).\end{cases}$

These are the two possible normal forms for a non-degenerate quadratic form
over $\mathbb{Z}_{2}$. Moreover, in both cases the symplectic and the
quadratic form are related by

(83)
$q(\bar{\mathsf{x}}\bar{\mathsf{y}})=q(\bar{\mathsf{x}})+q(\bar{\mathsf{y}})+(\bar{\mathsf{x}},\,\bar{\mathsf{y}})\quad\textrm{for
all }\bar{\mathsf{x}},\,\bar{\mathsf{y}}\in V.$

If $\phi\in\mathrm{Aut}(G)$, then $\phi$ induces a linear map
$\bar{\phi}\in\mathrm{End}(V)$; moreover, if $p=2$, then $\phi$ acts trivially
on $Z(G)=[G,\,G]\simeq\mathbb{Z}_{2}$, and this in turn implies that $\phi$
preserves the symplectic form on $V$. In other words, if we identify $V$ with
$(\mathbb{Z}_{2})^{2b}$ via the symplectic basis (81), we have
$\bar{\phi}\in\mathsf{Sp}(2b,\,\mathbb{Z}_{2})$.

We are now in a position to describe the structure of $\mathrm{Aut}(G)$, see
[Win72, Theorem 1].

######  Proposition 3.12.

Let $G$ be an extra-special group of order $2^{2b+1}$. Then the kernel of the
group homomorphism
$\mathrm{Aut}(G)\longrightarrow\mathsf{Sp}(2b,\,\mathbb{Z}_{2})$ given by
$\phi\mapsto\bar{\phi}$ is the subgroup $\mathrm{Inn}(G)$ of inner
automorphisms of $G$. Therefore
$\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)$ embeds in
$\mathsf{Sp}(2b,\,\mathbb{Z}_{2})$. More precisely, $\mathrm{Out}(G)$
coincides with the orthogonal group
$\mathsf{O}_{\epsilon}(2b,\,\mathbb{Z}_{2})$, of order

(84)
$|\mathsf{O}_{\epsilon}(2b,\,\mathbb{Z}_{2})|=2^{b(b-1)+1}(2^{b}-\epsilon)\prod_{i=1}^{b-1}(2^{2i}-1),$

associated with the quadratic form $\mathrm{\eqref{eq:form-of-quadratic-
forms}}$. Here $\epsilon=1$ if $G=\mathsf{H}_{2b+1}(\mathbb{Z}_{2})$ and
$\epsilon=-1$ if $G=\mathsf{G}_{2b+1}(\mathbb{Z}_{2})$.

######  Corollary 3.13.

Let $G$ be an extra-special group of order $2^{2b+1}$. We have

(85)
$|\mathrm{Aut}(G)|=2^{b(b+1)+1}(2^{b}-\epsilon)\prod_{i=1}^{b-1}(2^{2i}-1).$

###### Proof.

By Proposition 3.12 we get
$|\mathrm{Aut}(G)|=|\mathrm{Inn}(G)|\cdot|\mathsf{O}_{\epsilon}(2b,\,\mathbb{Z}_{2})|$.
Since $\mathrm{Inn}(G)\simeq G/Z(G)$ has order $2^{2b}$, the claim follows
from (84). ∎

In particular, plugging $b=2$ in (85), we can compute the orders of
automorphism groups of extra-special groups of order $32$, namely

(86)
$|\mathrm{Aut}(\mathsf{H}_{5}(\mathbb{Z}_{2}))|=1152,\quad|\mathrm{Aut}(\mathsf{G}_{5}(\mathbb{Z}_{2}))|=1920.$

Assume now that
$\mathfrak{S}=(\mathsf{r}_{11},\,\mathsf{t}_{11},\,\mathsf{r}_{12},\,\mathsf{t}_{12},\,\mathsf{r}_{21},\,\mathsf{t}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22},\mathsf{z})$
is a diagonal double Kodaira structure of type $(2,\,n)$ on an extra-special
group $G$ of order $32$; by Corollary 3.8, the element $\mathsf{z}$ is the
generator of $Z(G)\simeq\mathbb{Z}_{2}$, hence $n=2$. Then

(87)
$\bar{\mathfrak{S}}=(\mathsf{\bar{r}}_{11},\,\mathsf{\bar{t}}_{11},\,\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12},\,\mathsf{\bar{r}}_{21},\,\mathsf{\bar{t}}_{21},\,\mathsf{\bar{r}}_{22},\,\mathsf{\bar{t}}_{22})$

is an ordered set of generators for the symplectic $\mathbb{Z}_{2}$-vector
space $V=G/Z(G)\simeq(\mathbb{Z}_{2})^{4}$, and (77) yields the relations

(88)
$\begin{split}&(\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12})+(\mathsf{\bar{r}}_{11},\,\mathsf{\bar{t}}_{11})=1,\\\
&(\mathsf{\bar{r}}_{21},\,\mathsf{\bar{t}}_{21})+(\mathsf{\bar{r}}_{22},\,\mathsf{\bar{t}}_{22})=1,\\\
&(\mathsf{\bar{r}}_{1j},\,\mathsf{\bar{t}}_{2k})=\delta_{jk},\quad(\mathsf{\bar{r}}_{1j},\,\mathsf{\bar{r}}_{2k})=0\\\
&(\mathsf{\bar{t}}_{1j},\,\mathsf{\bar{r}}_{2k})=\delta_{jk},\quad(\mathsf{\bar{t}}_{1j},\,\mathsf{\bar{t}}_{2k})=0.\end{split}$

Conversely, given any set of generators $\bar{\mathfrak{S}}$ of $V$ as in
(87), whose elements satisfy (88), a diagonal double Kodaira structure of type
$(b,\,n)=(2,\,2)$ on $G$ inducing $\bar{\mathfrak{S}}$ is necessarily of the
form

(89)
$\mathfrak{S}=(\mathsf{r}_{11}\mathsf{z}^{a_{11}},\,\mathsf{t}_{11}\mathsf{z}^{b_{11}},\,\mathsf{r}_{12}\mathsf{z}^{a_{12}},\,\mathsf{t}_{12}\mathsf{z}^{b_{12}},\,\mathsf{r}_{21}\mathsf{z}^{a_{21}},\,\mathsf{t}_{21}\mathsf{z}^{b_{21}},\,\mathsf{r}_{22}\mathsf{z}^{a_{22}},\,\mathsf{t}_{22}\mathsf{z}^{b_{22}},\,\mathsf{z}),$

where $a_{ij},\,b_{ij}\in\\{0,\,1\\}$. This proves the following

######  Lemma 3.14.

The total number of diagonal double Kodaira structures of type
$(b,\,n)=(2,\,2)$ on an extra-special group $G$ of order $32$ is obtained
multiplying by $2^{8}$ the number of ordered sets of generators
$\bar{\mathfrak{S}}$ of $V$ as in $\eqref{eq:bar-Sigma}$, whose elements
satisfy (88). In particular, such a number does not depend on $G$.

We are now ready to state the main result of this section.

######  Theorem 3.15.

A finite group $G$ of order $32$ admits a diagonal double Kodaira structure if
and only if $G$ is extra-special. In this case, the following holds.

  * $\boldsymbol{(1)}$

Both extra-special groups of order $32$ admit $2211840=1152\cdot 1920$
distinct diagonal double Kodaira structures of type $(b,\,n)=(2,\,2)$. Every
such a structure $\mathfrak{S}$ is strong and satisfies
$\sigma(\mathfrak{S})=16$.

  * $\boldsymbol{(2)}$

If $G=G(32,\,49)=\mathsf{H}_{5}(\mathbb{Z}_{2})$, these structures form $1920$
orbits under the action of $\mathrm{Aut}(G)$.

  * $\boldsymbol{(3)}$

If $G=G(32,\,50)=\mathsf{G}_{5}(\mathbb{Z}_{2})$, these structures form $1152$
orbits under the action of $\mathrm{Aut}(G)$.

###### Proof.

We already know that non-extra-special groups of order $32$ admit no diagonal
double Kodaira structures (Proposition 3.11) and so, in the sequel, we can
assume that $G$ is extra-special.

Looking at the first two relations in (88), we see that we must consider four
cases:

  * $\boldsymbol{(a)}$

$\;(\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12})=0,\quad(\mathsf{\bar{r}}_{11},\,\mathsf{\bar{t}}_{11})=1,\quad(\mathsf{\bar{r}}_{21},\,\mathsf{\bar{t}}_{21})=0,\quad(\mathsf{\bar{r}}_{22},\,\mathsf{\bar{t}}_{22})=1,$

  * $\boldsymbol{(b)}$

$\;(\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12})=1,\quad(\mathsf{\bar{r}}_{11},\,\mathsf{\bar{t}}_{11})=0,\quad(\mathsf{\bar{r}}_{21},\,\mathsf{\bar{t}}_{21})=1,\quad(\mathsf{\bar{r}}_{22},\,\mathsf{\bar{t}}_{22})=0,$

  * $\boldsymbol{(c)}$

$\;(\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12})=0,\quad(\mathsf{\bar{r}}_{11},\,\mathsf{\bar{t}}_{11})=1,\quad(\mathsf{\bar{r}}_{21},\,\mathsf{\bar{t}}_{21})=1,\quad(\mathsf{\bar{r}}_{22},\,\mathsf{\bar{t}}_{22})=0,$

  * $\boldsymbol{(d)}$

$\;(\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12})=1,\quad(\mathsf{\bar{r}}_{11},\,\mathsf{\bar{t}}_{11})=0,\quad(\mathsf{\bar{r}}_{21},\,\mathsf{\bar{t}}_{21})=0,\quad(\mathsf{\bar{r}}_{22},\,\mathsf{\bar{t}}_{22})=1.$

Case $\boldsymbol{(a)}$. In this case the vectors
$\mathsf{\bar{r}}_{11},\,\mathsf{\bar{t}}_{11},\,\mathsf{\bar{r}}_{22},\,\mathsf{\bar{t}}_{22}$
are a symplectic basis of $V$, whereas the subspace
$W=\langle\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12},\,\mathsf{\bar{r}}_{21},\,\mathsf{\bar{t}}_{21}\rangle$
is isotropic, namely the symplectic form is identically zero on it. Since $V$
is a symplectic vector space of dimension $4$, the Witt index of $V$, i.e. the
dimension of a maximal isotropic subspace of $V$, is $\frac{1}{2}\dim(V)=2$,
see [Ar62, Théorèmes 3.10, 3.11]. On the other hand, we have
$(\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{22})=1$ and
$(\mathsf{\bar{t}}_{12},\,\,\mathsf{\bar{t}}_{22})=0$, hence
$\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12}$ are linearly independent and
so they must generate a maximal isotropic subspace; it follows that
$W=\langle\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12}\rangle$. Let us set
now

(90)
$\begin{split}(\mathsf{\bar{r}}_{11},\,\mathsf{\bar{r}}_{12})&=a,\quad(\mathsf{\bar{r}}_{11},\,\mathsf{\bar{t}}_{12})=b,\quad(\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{11})=c,\quad(\mathsf{\bar{t}}_{11},\,\mathsf{\bar{t}}_{12})=d,\\\
(\mathsf{\bar{r}}_{21},\,\mathsf{\bar{r}}_{22})&=e,\quad(\mathsf{\bar{r}}_{21},\,\mathsf{\bar{t}}_{22})=f,\quad(\mathsf{\bar{r}}_{22},\,\mathsf{\bar{t}}_{21})=g,\quad(\mathsf{\bar{t}}_{21},\,\mathsf{\bar{t}}_{22})=h,\\\
\end{split}$

where $a,\,b,\,c,\,d,\,e,\,f,\,g,\,h\in\mathbb{Z}_{2}$, and let us express the
remaining vectors of $\bar{\mathfrak{S}}$ in terms of the symplectic basis.
Standard computations yield

(91) $\displaystyle\mathsf{\bar{r}}_{12}$
$\displaystyle=c\mathsf{\bar{r}}_{11}+a\mathsf{\bar{t}}_{11}+\mathsf{\bar{r}}_{22},$
$\displaystyle\mathsf{\bar{t}}_{12}=d\mathsf{\bar{r}}_{11}+b\mathsf{\bar{t}}_{11}+\mathsf{\bar{t}}_{22},$
(92) $\displaystyle\mathsf{\bar{r}}_{21}$
$\displaystyle=\mathsf{\bar{r}}_{11}+f\mathsf{\bar{r}}_{22}+e\mathsf{\bar{t}}_{22},$
$\displaystyle\mathsf{\bar{t}}_{21}=\mathsf{\bar{t}}_{11}+h\mathsf{\bar{r}}_{22}+g\mathsf{\bar{t}}_{22}.$

Now recall that $W$ is isotropic; then, using the expressions in (91) and
imposing the relations

(93)
$\begin{array}[]{lll}(\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12})=0,&\quad(\mathsf{\bar{r}}_{12},\,\mathsf{\bar{r}}_{21})=0,&\quad(\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{21})=0,\\\
(\mathsf{\bar{r}}_{21},\,\mathsf{\bar{t}}_{12})=0,&\quad(\mathsf{\bar{t}}_{12},\,\mathsf{\bar{t}}_{21})=0,&\quad(\mathsf{\bar{r}}_{21},\,\mathsf{\bar{t}}_{21})=0,\end{array}$

we get

(94) $\begin{array}[]{lll}ad+bc=1,&\quad a+e=0,&\quad\quad c+g=0,\\\ \quad
b+f=0,&\quad d+h=0,&\quad eh+fg=1.\end{array}$

Summing up, the elements $\mathsf{\bar{r}}_{12}$, $\mathsf{\bar{t}}_{12}$,
$\mathsf{\bar{r}}_{21}$, $\mathsf{\bar{t}}_{21}$ can be determined from the
symplectic basis via the relations

(95) $\displaystyle\mathsf{\bar{r}}_{12}$
$\displaystyle=c\mathsf{\bar{r}}_{11}+a\mathsf{\bar{t}}_{11}+\mathsf{\bar{r}}_{22},$
$\displaystyle\mathsf{\bar{t}}_{12}=d\mathsf{\bar{r}}_{11}+b\mathsf{\bar{t}}_{11}+\mathsf{\bar{t}}_{22},$
(96) $\displaystyle\mathsf{\bar{r}}_{21}$
$\displaystyle=\mathsf{\bar{r}}_{11}+b\mathsf{\bar{r}}_{22}+a\mathsf{\bar{t}}_{22},$
$\displaystyle\mathsf{\bar{t}}_{21}=\mathsf{\bar{t}}_{11}+d\mathsf{\bar{r}}_{22}+c\mathsf{\bar{t}}_{22},$

where $a,\,b,\,c,\,d\in\mathbb{Z}_{2}$ and $ad+bc=1$. Conversely, given any
symplectic basis
$\mathsf{\bar{r}}_{11},\,\mathsf{\bar{t}}_{11},\,\mathsf{\bar{r}}_{22},\,\mathsf{\bar{t}}_{22}$
of $V$ and elements $\mathsf{\bar{r}}_{12}$, $\mathsf{\bar{t}}_{12}$,
$\mathsf{\bar{r}}_{21}$, $\mathsf{\bar{t}}_{21}$ as in (95), with $ad+bc=1$,
we get a set of generators $\bar{\mathfrak{S}}$ of $V$ having the form (87),
and whose elements satisfy (88). Thus, the total number of such
$\bar{\mathfrak{S}}$ in this case is given by

(97)
$|\mathsf{Sp}(4,\,\mathbb{Z}_{2})|\cdot|\mathsf{GL}(2,\,\mathbb{Z}_{2})|=720\cdot
6=4320$

and so, by Lemma 3.14, the corresponding number of diagonal double Kodaira
structures is $2^{8}\cdot 4320=1105920$. All these structures are strong: in
fact, we have

(98)
$\begin{split}K_{1}=\langle\mathsf{r}_{11},\,\mathsf{t}_{11},\,\mathsf{r}_{12},\,\mathsf{t}_{12}\rangle&=\langle\mathsf{r}_{11},\,\mathsf{t}_{11},\,\mathsf{r}_{11}^{c}\mathsf{t}_{11}^{a}\mathsf{r}_{22},\,\mathsf{r}_{11}^{d}\mathsf{t}_{11}^{b}\mathsf{t}_{22}\rangle\\\
&=\langle\mathsf{r}_{11},\,\mathsf{t}_{11},\,\mathsf{r}_{22},\,\mathsf{t}_{22}\rangle=G\end{split}$
(99)
$\begin{split}K_{2}=\langle\mathsf{r}_{21},\,\mathsf{t}_{21},\,\mathsf{r}_{22},\,\mathsf{t}_{22}\rangle&=\langle\mathsf{r}_{11}\mathsf{r}_{22}^{b}\mathsf{t}_{22}^{a},\,\mathsf{t}_{11}\mathsf{r}_{22}^{d}\mathsf{t}_{22}^{c},\,\mathsf{r}_{22},\,\mathsf{t}_{22}\rangle\\\
&=\langle\mathsf{r}_{11},\,\mathsf{t}_{11},\,\mathsf{r}_{22},\,\mathsf{t}_{22}\rangle=G,\end{split}$

the last equality following in both cases because
$\langle\mathsf{\bar{r}}_{11},\,\mathsf{\bar{t}}_{11},\,\mathsf{\bar{r}}_{22},\,\mathsf{\bar{t}}_{22}\rangle=V$
and $[\mathsf{r}_{11},\,\mathsf{t}_{11}]=\mathsf{z}$.

Case $\boldsymbol{(b)}$. In this situation, the elements
$\\{\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12},\,\mathsf{\bar{r}}_{21},\,\mathsf{\bar{t}}_{21}\\}$
form a symplectic basis for $V$, whereas
$W=\langle\mathsf{\bar{r}}_{11},\,\mathsf{\bar{t}}_{11},\,\mathsf{\bar{r}}_{22},\,\mathsf{\bar{t}}_{22}\rangle$
is an isotropic subspace. The same calculations as in case $(a)$ show that
there are again $1105920$ diagonal double Kodaira structures.

Case $\boldsymbol{(c)}$. This case do not occur. In fact, in this situation
the subspace
$W=\langle\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12},\,\mathsf{\bar{r}}_{21}\rangle$
is isotropic. Take a linear combination of its generators giving the zero
vector, namely

(100)
$a\mathsf{\bar{r}}_{12}+b\mathsf{\bar{t}}_{12}+c\mathsf{\bar{r}}_{21}=0.$

Pairing with $\mathsf{\bar{t}}_{21}$, $\mathsf{\bar{t}}_{22}$,
$\mathsf{\bar{r}}_{22}$, we get $c=a=b=0$. Thus,
$\mathsf{\bar{r}}_{12},\,\mathsf{\bar{t}}_{12},\,\mathsf{\bar{r}}_{21}$ are
linearly independent, and $W$ is an isotropic subspace of dimension $3$ inside
the $4$-dimensional symplectic space $V$, contradiction.

Case $\boldsymbol{(d)}$. This case is obtained from $(c)$ by exchanging the
indices $1$ and $2$, so it does not occur, either.

Summarizing, we have found $1105920$ diagonal double Kodaira structures in
cases $(a)$ and $(b)$, and no structure at all in cases $(c)$ and $(d)$. So
the total number of diagonal double Kodaira structures on $G$ is $2211840$,
and this concludes the proof of part $\boldsymbol{(1)}$.

Now observe that, since every diagonal double Kodaira structure $\mathfrak{S}$
generates $G$, the only automorphism $\phi$ of $G$ fixing $\mathfrak{S}$
elementwise is the identity. This means that $\mathrm{Aut}(G)$ acts freely on
the set of diagonal double Kodaira structures, hence the number of orbits is
obtained dividing $2211840$ by $|\mathrm{Aut}(G)|$. Parts $\boldsymbol{(2)}$
and $\boldsymbol{(3)}$ now follow from (86), and we are done. ∎

######  Example 3.16.

Let us give an explicit example of diagonal double Kodaira structure on an
extra-special group $G$ of order $32$, by using the construction described in
the proof of part $\mathbf{(1)}$ of Theorem 3.15. Referring to the
presentations for $\mathsf{H}_{5}(\mathbb{Z}_{2})$ and
$\mathsf{G}_{5}(\mathbb{Z}_{2})$ given in Proposition 1.9, we start by
choosing in both cases the following elements, whose images give a symplectic
basis for $V$:

(101)
$\mathsf{r}_{11}=\mathsf{r}_{1},\quad\mathsf{t}_{11}=\mathsf{t}_{1},\quad\mathsf{r}_{22}=\mathsf{r}_{2},\quad\mathsf{t}_{22}=\mathsf{t}_{2}.$

Choosing $a=d=1$ and $b=c=0$ in (95), we find the remaining elements,
obtaining the diagonal double Kodaira structure

(102) $\displaystyle\mathsf{r}_{11}$ $\displaystyle=\mathsf{r}_{1},$
$\displaystyle\quad\mathsf{t}_{11}$ $\displaystyle=\mathsf{t}_{1},$
$\displaystyle\quad\mathsf{r}_{12}$
$\displaystyle=\mathsf{r}_{2}\,\mathsf{t}_{1},$
$\displaystyle\quad\mathsf{t}_{12}$
$\displaystyle=\mathsf{r}_{1}\,\mathsf{t}_{2}$ $\displaystyle\mathsf{r}_{21}$
$\displaystyle=\mathsf{r}_{1}\,\mathsf{t}_{2},$
$\displaystyle\quad\mathsf{t}_{21}$
$\displaystyle=\mathsf{r}_{2}\,\mathsf{t}_{1},$
$\displaystyle\quad\mathsf{r}_{22}$ $\displaystyle=\mathsf{r}_{2},$
$\displaystyle\quad\mathsf{t}_{22}$ $\displaystyle=\mathsf{t}_{2}.$

######  Remark 3.17.

Theorem 3.15 should be compared with previous results of [CaPol19] and
[Pol20], regarding the construction of diagonal double Kodaira structures on
some extra-special groups of order at least $2^{7}=128$. The examples on
extra-special groups of order $32$ presented here are really new, in the sense
that they cannot be obtained by taking the image of structures on extra-
special groups of bigger order: in fact, an extra-special group admits no non-
abelian proper quotients, cf. Example 3.6.

######  Remark 3.18.

Although it is known that the pure braid group $\mathsf{P}_{2}(\Sigma_{b})$ is
residually $p$-finite for all prime number $p\geq 2$, see [BarBel09, pp.
1481-1490], it is a non-trivial problem to understand its non-abelian, finite
quotients. The extra-special examples of order at least $128$ cited in Remark
3.17 were the outcome of the first (as far as we know) systematic
investigation of this matter. Our approach in the present work sheds some new
light on this problem, providing a sharp lower bound for the order of a non-
abelian quotient $G$ of $\mathsf{P}_{2}(\Sigma_{b})$, under the assumption
that the quotient map $\varphi\colon\mathsf{P}_{2}(\Sigma_{b})\longrightarrow
G$ does not factor through
$\pi_{1}(\Sigma_{b}\times\Sigma_{b},\,\mathscr{P})$; indeed, by [CaPol19,
Remark 1.7 (iv)], the factorization occurs if and only if $\varphi(A_{12})$ is
trivial. More precisely, we showed that, for every such a quotient, the
inequality $|G|\geq 32$ holds, with equality if and only if $G$ is extra-
special: in fact, both extra-special groups of order $32$ do appear as
quotients of $\mathsf{P}_{2}(\Sigma_{2})$. Moreover, for these groups, Theorem
3.15 also computes the number of distinct group epimorphisms
$\varphi\colon\mathsf{P}_{2}(\Sigma_{2})\longrightarrow G$ such that
$\varphi(A_{12})=\mathsf{z}$, and the number of their equivalence classes up
to the natural action of $\operatorname{Aut}(G)$.

##  4\. Geometrical application: diagonal double Kodaira fibrations

Recall that a _Kodaira fibration_ is a smooth, connected holomorphic fibration
$f_{1}\colon S\longrightarrow B_{1}$, where $S$ is a compact complex surface
and $B_{1}$ is a compact complex curve, which is not isotrivial. The genus
$b_{1}:=g(B_{1})$ is called the _base genus_ of the fibration, whereas the
genus $g:=g(F)$, where $F$ is any fibre, is called the _fibre genus_.

######  Definition 4.1.

A _double Kodaira surface_ is a compact complex surface $S$, endowed with a
_double Kodaira fibration_ , namely a surjective, holomorphic map $f\colon
S\longrightarrow B_{1}\times B_{2}$ yielding, by composition with the natural
projections, two Kodaira fibrations $f_{i}\colon S\longrightarrow B_{i}$,
$i=1,\,2$.

The aim of this section is to show how the existence of diagonal double
Kodaira structures is equivalent to the existence of some special double
Kodaira fibrations, that we call _diagonal double Kodaira fibrations_. We
closely follow the treatment given in [Pol20, Section 3].

With a slight abuse of notation, in the sequel we will use the symbol
$\Sigma_{b}$ to indicate both a smooth complex curve of genus $b$ and its
underlying real surface. By Grauert-Remmert’s extension theorem and Serre’s
GAGA, the group epimorphism
$\varphi\colon\mathsf{P}_{2}(\Sigma_{b})\longrightarrow G$ described in
Proposition 2.6 yields the existence of a smooth, complex, projective surface
$S$ endowed with a Galois cover

(103) $\mathbf{f}\colon S\longrightarrow\Sigma_{b}\times\Sigma_{b},$

with Galois group $G$ and branched precisely over $\Delta$ with branching
order $n$, see [CaPol19, Proposition 3.4]. Composing the left homomorphism in
(68) with $\varphi\colon\mathsf{P}_{2}(\Sigma_{b})\longrightarrow G$, we get
two homomorphisms

(104) $\varphi_{1}\colon\pi_{1}(\Sigma_{b}-\\{p_{2}\\},\,p_{1})\longrightarrow
G,\quad\varphi_{2}\colon\pi_{1}(\Sigma_{b}-\\{p_{1}\\},\,p_{2})\longrightarrow
G,$

whose respective images coincide with the subgroups $K_{1}$ and $K_{2}$
defined in (72). By construction, these are the homomorphisms induced by the
restrictions $\mathbf{f}_{i}\colon\Gamma_{i}\longrightarrow\Sigma_{b}$ of the
Galois cover $\mathbf{f}\colon S\longrightarrow\Sigma_{b}\times\Sigma_{b}$ to
the fibres of the two natural projections
$\pi_{i}\colon\Sigma_{b}\times\Sigma_{b}\longrightarrow\Sigma_{b}$. Since
$\Delta$ intersects transversally at a single point all the fibres of the
natural projections, it follows that both such restrictions are branched at
precisely one point, and the number of connected components of the smooth
curve $\Gamma_{i}\subset S$ equals the index $m_{i}:=[G:K_{i}]$ of $K_{i}$ in
$G$.

So, taking the Stein factorizations of the compositions
$\pi_{i}\circ\mathbf{f}\colon S\longrightarrow\Sigma_{b}$ as in the diagram
below

(105)
${S}$${\Sigma_{b}}$${\Sigma_{b_{i}}}$$\scriptstyle{\pi_{i}\circ\mathbf{f}}$$\scriptstyle{f_{i}}$$\scriptstyle{\theta_{i}}$

we obtain two distinct Kodaira fibrations $f_{i}\colon
S\longrightarrow\Sigma_{b_{i}}$, hence a double Kodaira fibration by
considering the product morphism

(106) $f=f_{1}\times f_{2}\colon
S\longrightarrow\Sigma_{b_{1}}\times\Sigma_{b_{2}}.$

######  Definition 4.2.

We call $f\colon S\longrightarrow\Sigma_{b_{1}}\times\Sigma_{b_{2}}$ the
_diagonal double Kodaira fibration_ associated with the diagonal double
Kodaira structure $\mathfrak{S}$ on the finite group $G$. Conversely, we will
say that a double Kodaira fibration $f\colon
S\longrightarrow\Sigma_{b_{1}}\times\Sigma_{b_{2}}$ is _of diagonal type_
$(b,\,n)$ if there exists a finite group $G$ and a diagonal double Kodaira
structure $\mathfrak{S}$ of type $(b,\,n)$ on it such that $f$ is associated
with $\mathfrak{S}$.

Since the morphism $\theta_{i}\colon\Sigma_{b_{i}}\longrightarrow\Sigma_{b}$
is étale of degree $m_{i}$, by using the Hurwitz formula we obtain

(107) $b_{1}-1=m_{1}(b-1),\quad b_{2}-1=m_{2}(b-1).$

Moreover, the fibre genera $g_{1}$, $g_{2}$ of the Kodaira fibrations
$f_{1}\colon S\longrightarrow\Sigma_{b_{1}}$, $f_{2}\colon
S\longrightarrow\Sigma_{b_{2}}$ are computed by the formulae

(108) $2g_{1}-2=\frac{|G|}{m_{1}}(2b-2+\mathfrak{n}),\quad
2g_{2}-2=\frac{|G|}{m_{2}}\left(2b-2+\mathfrak{n}\right),$

where $\mathfrak{n}:=1-1/n$. Finally, the surface $S$ fits into a diagram

(109)
${S}$${\Sigma_{b}\times\Sigma_{b}}$${\Sigma_{b_{1}}\times\Sigma_{b_{2}}}$$\scriptstyle{\mathbf{f}}$$\scriptstyle{f}$$\scriptstyle{\theta_{1}\times\theta_{2}}$

so that the diagonal double Kodaira fibration $f\colon
S\longrightarrow\Sigma_{b_{1}}\times\Sigma_{b_{2}}$ is a finite cover of
degree $\frac{|G|}{m_{1}m_{2}}$, branched precisely over the curve

(110)
$(\theta_{1}\times\theta_{2})^{-1}(\Delta)=\Sigma_{b_{1}}\times_{\Sigma_{b}}\Sigma_{b_{2}}.$

Such a curve is always smooth, being the preimage of a smooth divisor via an
étale morphism. However, it is reducible in general, see [CaPol19, Proposition
3.11]. The invariants of $S$ can be now computed as follows, see [CaPol19,
Proposition 3.8].

######  Proposition 4.3.

Let $f\colon S\longrightarrow\Sigma_{b_{1}}\times\Sigma_{b_{2}}$ be a diagonal
double Kodaira fibration, associated with a diagonal double Kodaira structure
$\mathfrak{S}$ of type $(b,\,n)$ on a finite group $G$. Then we have

(111)
$\begin{split}c_{1}^{2}(S)&=|G|\,(2b-2)(4b-4+4\mathfrak{n}-\mathfrak{n}^{2})\\\
c_{2}(S)&=|G|\,(2b-2)(2b-2+\mathfrak{n})\end{split}$

where $\mathfrak{n}=1-1/n$. As a consequence, the slope and the signature of
$S$ can be expressed as

(112)
$\begin{split}\nu(S)&=\frac{c_{1}^{2}(S)}{c_{2}(S)}=2+\frac{2\mathfrak{n}-\mathfrak{n}^{2}}{2b-2+\mathfrak{n}}\\\
\sigma(S)&=\frac{1}{3}\left(c_{1}^{2}(S)-2c_{2}(S)\right)=\frac{1}{3}\,|G|\,(2b-2)\left(1-\frac{1}{n^{2}}\right)=\sigma(\mathfrak{S})\end{split}$

######  Remark 4.4.

By definition, the diagonal double Kodaira structure $\mathfrak{S}$ is strong
if and only if $m_{1}=m_{2}=1$, that in turn implies $b_{1}=b_{2}=b$, i.e.,
$f=\mathbf{f}$. In other words, $\mathfrak{S}$ is strong if and only if no
Stein factorization as in (105) is needed or, equivalently, if and only if the
Galois cover $\mathbf{f}\colon S\longrightarrow\Sigma_{b}\times\Sigma_{b}$
induced by (66) is already a double Kodaira fibration, branched on the
diagonal $\Delta\subset\Sigma_{b}\times\Sigma_{b}$.

######  Remark 4.5.

Every Kodaira fibred surface $S$ satisfies $\sigma(S)>0$, see the introduction
to [LLR20]; moreover, since $S$ is a differentiable $4$-manifold that is a
real surface bundle, its signature is divisible by $4$, see [Mey73]. In
addition, if $S$ is associated with a diagonal double Kodaira structure of
type $(b,\,n)$, with $n$ odd, then $K_{S}$ is $2$-divisible in
$\textrm{Pic}(S)$ and so $\sigma(S)$ is a positive multiple of $16$ by
Rokhlin’s theorem, see [CaPol19, Remark 3.9].

######  Remark 4.6.

Not all double Kodaira fibration are of diagonal type. In fact, if $S$ is of
diagonal type then its slope satisfies $\nu(S)=2+s$, where $s$ is rational and
$0<s<6-4\sqrt{2}$, see [Pol20, Proposition 3.12 and Remark 3.13].

We are now ready to give a geometric interpretation of Proposition 3.9,
Proposition 3.11 and Theorem 3.15 in terms of double Kodaira fibrations.

######  Theorem 4.7.

Let $G$ be a finite group and

(113) $\mathbf{f}\colon S\longrightarrow\Sigma_{b}\times\Sigma_{b}$

be a Galois cover with Galois group $G$, branched over the diagonal $\Delta$
with branching order $n\geq 2$. Then the following hold.

  * $\boldsymbol{(1)}$

We have $|G|\geq 32$, with equality precisely when $G$ is extra-special.

  * $\boldsymbol{(2)}$

If $G=G(32,\,49)=\mathsf{H}_{5}(\mathbb{Z}_{2})$ and $b=2$, there are $1920$
$G$-covers of type (113), up to cover isomorphisms.

  * $\boldsymbol{(3)}$

If $G=G(32,\,50)=\mathsf{G}_{5}(\mathbb{Z}_{2})$ and $b=2$, there are $1152$
$G$-covers of type (113), up to cover isomorphisms.

Finally, in both cases $\boldsymbol{(2)}$ and $\boldsymbol{(3)}$, we have
$n=2$ and each cover $\mathbf{f}$ is a double Kodaira fibration with

(114) $b_{1}=b_{2}=2,\quad g_{1}=g_{2}=41,\quad\sigma(S)=16.$

###### Proof.

By the result of Section 4, a cover as in (113), branched over $\Delta$ with
order $n$, exists if and only if $G$ admits a double Kodaira structure of type
$(b,\,n)$, and the number of such covers, up to cover isomorphisms, equals the
number of structures up the natural action of $\mathrm{Aut}(G)$. Then,
$\boldsymbol{(1)}$, $\boldsymbol{(2)}$ and $\boldsymbol{(3)}$ can be deduced
from the corresponding statements in Theorem 3.15. The same theorem tells us
that all double Kodaira structures on an extra-special group of order $32$ are
strong, hence the cover $\mathbf{f}$ is already a double Kodaira fibration and
no Stein factorization is needed (Remark 4.4). The fibre genera, the slope and
the signature of $S$ can be now computed by using (108) and (112). ∎

As a consequence, we obtain a sharp lower bound for the signature of a
diagonal double Kodaira fibration or, equivalently, of a diagonal double
Kodaira structure.

######  Corollary 4.8.

Let $f\colon S\longrightarrow\Sigma_{b_{1}}\times\Sigma_{b_{2}}$ be a diagonal
double Kodaira fibration, associated with a diagonal double Kodaira structure
of type $(b,\,n)$ on a finite group $G$. Then $\sigma(S)\geq 16$, and equality
holds precisely when $(b,\,n)=(2,\,2)$ and $G$ is an extra-special group of
order $32$.

###### Proof.

Theorem 3.15 implies $|G|\geq 32$. Since $b\geq 2$ and $n\geq 2$, from (112)
we get

(115) $\sigma(S)=\frac{1}{3}\,|G|\,(2b-2)\left(1-\frac{1}{\;n^{2}}\right)\\\
\geq\frac{1}{3}\cdot 32\cdot(2\cdot 2-2)\left(1-\frac{1}{\;2^{2}}\right)=16,$

and equality holds if and only if we are in the situation described in Theorem
4.7, namely, $b=n=2$ and $G$ an extra-special group of order $32$. ∎

These results provide, in particular, new “double solutions” to a problem,
posed by G. Mess, from Kirby’s problem list in low-dimensional topology
[Kir97, Problem 2.18 A], asking what is the smallest number $b$ for which
there exists a real surface bundle over a real surface with base genus $b$ and
non-zero signature. We actually have $b=2$, also for double Kodaira
fibrations, as shown in [CaPol19, Proposition 3.19] and [Pol20, Theorem 3.6]
by using double Kodaira structures of type $(2,\,3)$ on extra-special groups
of order $3^{5}$. Those fibrations had signature $144$ and fibre genera $325$;
by using our new examples, we are now able to substantially lower both these
values.

######  Theorem 4.9.

Let $S$ be the diagonal double Kodaira surface associated with a strong
diagonal double Kodaira structure of type $(b,\,n)=(2,\,2)$ on an extra-
special group $G$ of order $32$. Then the real manifold $M$ underlying $S$ is
a closed, orientable $4$-manifold of signature $16$ that can be realized as a
real surface bundle over a real surface of genus $2$, with fibre genus $41$,
in two different ways.

Theorem 4.7 also implies the following partial answer to [CaPol19, Question
3.20].

######  Corollary 4.10.

Let $g_{\mathrm{min}}$ and $\sigma_{\mathrm{min}}$ be the minimal possible
fibre genus and signature for a double Kodaira fibration $f\colon
S\longrightarrow\Sigma_{2}\times\Sigma_{2}$. Then we have

(116) $g_{\mathrm{min}}\leq 41,\quad\sigma_{\mathrm{min}}\leq 16.$

In fact, it is an interesting question whether $16$ and $41$ are the minimum
possible values for the signature and the fibre genus of a (non necessarily
diagonal) double Kodaira fibration $f\colon
S\longrightarrow\Sigma_{2}\times\Sigma_{2}$, but we will not address this
problem here.

######  Remark 4.11.

Constructing (double) Kodaira fibrations with small signature is a rather
difficult problem. As far as we know, before our work the only examples with
signature $16$ were the ones listed in [LLR20, Table 3, Cases 6.2, 6.6, 6.7
(Type 1), 6.9]. The examples provided by Theorem 4.7 are new, since both the
base genera and the fibre genera are different. Note that our results also
show that _every_ curve of genus $2$ (and not only some special curve with
extra automorphisms) is the base of a double Kodaira fibration with signature
$16$. Thus, we obtain two families of dimension $3$ of such fibrations that,
to the best of our knowledge, provide the first examples of a positive-
dimensional families of double Kodaira fibrations with small signature.

######  Remark 4.12.

Let $f\colon S\longrightarrow\Sigma_{b}\times\Sigma_{b}$ be a double Kodaira
fibration, associated with a strong diagonal double Kodaira structure of type
$(b,\,n)$ on a finite group $G$. Then the branch locus of $f$ is
$\Delta\subset\Sigma_{b}\times\Sigma_{b}$, namely the graph of the identity
$\operatorname{id}\colon\Sigma_{b}\longrightarrow\Sigma_{b}$, and so, adopting
the terminology in [CatRol09], we say that $f$ is _very simple_. Let us denote
by $\mathfrak{M}_{S}$ the connected component of the Gieseker moduli space of
surfaces of general type containing the class of $S$, and by $\mathcal{M}_{b}$
the moduli space of smooth curves of genus $b$. Thus, by applying [Rol10, Thm.
1.7], we infer that every surface in $\mathfrak{M}_{S}$ is still a very simple
double Kodaira fibration and that there is a natural map of schemes

(117) $\mathcal{M}_{b}\longrightarrow\mathfrak{M}_{S},$

which is an isomorphism on geometric points. Roughly speaking, since the
branch locus $\Delta\subset\Sigma_{b}\times\Sigma_{b}$ is rigid, all the
deformations of $S$ are realized by deformations of
$\Sigma_{b}\times\Sigma_{b}$ preserving the diagonal, hence by deformations of
$\Sigma_{b}$, cf. [CaPol19, Proposition 3.22]. In particular, this shows that
$\mathfrak{M}_{S}$ is a connected and irreducible component of the Gieseker
moduli space.

###  4.1. The computation of $H_{1}(S,\,\mathbb{Z})$

We end this section by computing the first homology group
$H_{1}(S,\,\mathbb{Z})$, where $f\colon
S\longrightarrow\Sigma_{2}\times\Sigma_{2}$ is the diagonal double Kodaira
fibration associated with a diagonal double Kodaira structure of type
$(b,\,n)=(2,\,2)$ on an extra-special group of order $32$. To this purpose, we
will make use of the following result, which is a consequence of [Fox57,
Theorem p. 254].

######  Proposition 4.13.

Let $G$ be a finite group and
$\varphi\colon\mathsf{P}_{2}(\Sigma_{b})\longrightarrow G$ be a surjective
group homomorphism, such that $\varphi(A_{12})$ has order $n\geq 2$. If
$f\colon S\longrightarrow\Sigma_{b_{1}}\times\Sigma_{b_{2}}$ is the diagonal
double Kodaira fibration associated with $\varphi$, then $\pi_{1}(S)$ sits
into a short exact sequence

(118)
$1\longrightarrow\pi_{1}(S)\longrightarrow\mathsf{P}_{2}(\Sigma_{b})^{\operatorname{orb}}\stackrel{{\scriptstyle\,\,\,\varphi^{\operatorname{orb}}}}{{\longrightarrow}}G\longrightarrow
1,$

where $\mathsf{P}_{2}(\Sigma_{b})^{\operatorname{orb}}$ is the quotient of
$\mathsf{P}_{2}(\Sigma_{b})$ by the normal closure of the cyclic subgroup
$\langle A_{12}^{n}\rangle$, and
$\varphi^{\operatorname{orb}}\colon\mathsf{P}_{2}(\Sigma_{b})^{\operatorname{orb}}\longrightarrow
G$ is the group epimorphism naturally induced by $\varphi$.

Proposition 4.13 allows one, at least in principle, to compute $\pi_{1}(S)$,
and so its abelianization $H_{1}(S,\,\mathbb{Z})$. However, doing all the
calculations by hand seems quite difficult, so we resorted to the Computer
Algebra System `GAP4`, see [GAP4]. The reader can find the idea behind the
calculation and the corresponding script in Appendix B, while here we just
report the result.

######  Proposition 4.14.

Let $f\colon S\longrightarrow\Sigma_{2}\times\Sigma_{2}$ be the diagonal
double Kodaira fibration associated with a diagonal double Kodaira structure
of type $(b,\,n)=(2,\,2)$ on an extra-special group $G$ of order $32$. Then

(119) $H_{1}(S,\,\mathbb{Z})=\mathbb{Z}^{8}\oplus(\mathbb{Z}_{2})^{4}.$

In particular, this homology group is independent both on $G$ and on the
chosen structure on it.

######  Corollary 4.15.

Let $f\colon S\longrightarrow\Sigma_{2}\times\Sigma_{2}$ be the diagonal
double Kodaira fibration associated with a diagonal double Kodaira structure
of type $(b,\,n)=(2,\,2)$ on an extra-special group of order $32$. Then

(120) $c_{1}^{2}(S)=368,\quad c_{2}(S)=160,\quad p_{g}(S)=47,\quad q(S)=4.$

###### Proof.

The integers $c_{1}^{2}(S),\,c_{2}(S)$ can be computed by using (111). Since
$\mathsf{b}_{1}(S)=8$ (Proposition 4.14), it follows $q(S)=4$. On the other
hand, Noether formula gives

(121)
$1-q(S)+p_{g}(S)=\chi(\mathcal{O}_{S})=\frac{c_{1}^{2}(S)+c_{2}(S)}{12}=44,$

hence $p_{g}(S)=47$. ∎

Finally, we want to relate Proposition 4.14 to some facts about monodromy.
Recall that, by [CatRol09, Proposition 2.5], given a Kodaira fibration
$S\longrightarrow\Sigma_{b}$, with base genus $b$ and fibre genus $g$, there
is a short exact sequence of fundamental groups

(122)
$1\longrightarrow\pi_{1}(\Sigma_{g})\longrightarrow\pi_{1}(S)\longrightarrow\pi_{1}(\Sigma_{b})\longrightarrow
1,$

which induces, by conjugation, a monodromy representation
$\pi_{1}(\Sigma_{b})\longrightarrow\operatorname{Out}(\pi_{1}(\Sigma_{g}))$.
Taking the abelianization of the right side, and dualizing over $\mathbb{Q}$,
we obtain a monodromy representation

(123)
$\rho_{\pi_{1}(\Sigma_{b})}\colon\pi_{1}(\Sigma_{b})\longrightarrow\operatorname{Aut}(H^{1}(\Sigma_{g},\,\mathbb{Q})),$

whose invariant subspace will be denoted by
$H^{1}(\Sigma_{g},\,\mathbb{Q})^{\pi_{1}(\Sigma_{b})}$.

Now, let $f\colon S\longrightarrow\Sigma_{b}\times\Sigma_{b}$ be a diagonal
double Kodaira fibration, associated with a group epimorphism
$\varphi\colon\mathsf{P}_{2}(\Sigma_{b})\longrightarrow G$ such that
$\varphi(A_{12})$ has order $n\geq 2$. Set $\\{i,\,j\\}=\\{1,\,2\\}$ and let
$f_{i}\colon S\longrightarrow\Sigma_{b}$ be the Kodaira fibration obtained
composing $f$ with the natural projection
$\pi_{i}\colon\Sigma_{b}\times\Sigma_{b}\longrightarrow\Sigma_{b}$ onto the
$i$th factor. Assume moreover that the induced group homomorphism
$\varphi_{j}\colon\pi_{1}(\Sigma_{b}-\\{p_{i}\\})\longrightarrow G$ is
surjective. Then $\pi_{1}(\Sigma_{g})$ fits into a short exact sequence

(124)
$1\longrightarrow\pi_{1}(\Sigma_{g})\longrightarrow\pi_{1}(\Sigma_{b}-\\{p_{i}\\})^{\operatorname{orb}}\stackrel{{\scriptstyle\,\,\,\varphi_{j}^{\operatorname{orb}}}}{{\longrightarrow}}G\longrightarrow
1,$

where

$\pi_{1}(\Sigma_{b}-\\{p_{i}\\})^{\operatorname{orb}}=\big{\langle}\rho_{j1},\,\tau_{j1},\ldots,\rho_{jb},\,\tau_{jb},\;A_{12}\mid
A_{12}\prod_{t=1}^{b}[\rho_{jt},\,\tau_{jt}]=1,\,A_{12}^{n}=1\big{\rangle},$

see (70), and
$\varphi_{j}^{\operatorname{orb}}\colon\pi_{1}(\Sigma_{b}-\\{p_{i}\\})^{\operatorname{orb}}\longrightarrow
G$ is the group epimorphism naturally induced by $\varphi_{j}$.
Correspondingly, we have a monodromy representation

(125) $\rho_{G}\colon
G\longrightarrow\operatorname{Aut}(H^{1}(\Sigma_{g},\,\mathbb{Q})),$

whose invariant subspace will be denoted by
$H^{1}(\Sigma_{g},\,\mathbb{Q})^{G}$.

######  Proposition 4.16.

Let $f\colon S\longrightarrow\Sigma_{b}\times\Sigma_{b}$ be a diagonal double
Kodaira fibration as above. Then the following holds.

  * $\boldsymbol{(1)}$

$\dim H^{1}(S,\,\mathbb{Q})=\dim
H^{1}(\Sigma_{g},\,\mathbb{Q})^{\pi_{1}(\Sigma_{b})}+2b$

  * $\boldsymbol{(2)}$

$H^{1}(\Sigma_{g},\,\mathbb{Q})^{G}\subseteq
H^{1}(\Sigma_{g},\,\mathbb{Q})^{\pi_{1}(\Sigma_{b})}$

###### Proof.

$\boldsymbol{(1)}$ If $\iota\colon\Sigma_{g}\longrightarrow S$ is the
inclusion, the Hochschild-Serre spectral sequence in cohomology associated
with (122) provides a short exact sequence of $\mathbb{Q}$-vector spaces

(126) $0\longrightarrow H^{1}(\Sigma_{b},\,\mathbb{Q})\stackrel{{\scriptstyle
f_{i}^{*}}}{{\longrightarrow}}H^{1}(S,\,\mathbb{Q})\stackrel{{\scriptstyle\iota^{*}}}{{\longrightarrow}}H^{1}(\Sigma_{g},\,\mathbb{Q})^{\pi_{1}(\Sigma_{b})}\longrightarrow
0,$

see for instance [Breg18, p. 5] or [Vid19, p. 740], so the claim follows.

$\boldsymbol{(2)}$ The $G$-cover $h\colon\Sigma_{g}\longrightarrow\Sigma_{b}$
define an injective pull-back map $h^{*}\colon
H^{1}(\Sigma_{b},\,\mathbb{Q})\longrightarrow H^{1}(\Sigma_{g},\,\mathbb{Q})$,
whose image is precisely $H^{1}(\Sigma_{g},\,\mathbb{Q})^{G}$. So it suffices
to check that $h^{*}$ is invariant under the monodromy map
$\rho_{\pi_{1}(\Sigma_{b})}$ and, to this purpose, we consider the
factorization of $h$ given as follows:

(127)
$\Sigma_{g}\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}S\stackrel{{\scriptstyle
f}}{{\longrightarrow}}\Sigma_{b}\times\Sigma_{b}\stackrel{{\scriptstyle\pi_{i}}}{{\longrightarrow}}\Sigma_{b}.$

By (126), the image of $\iota^{*}\colon H^{1}(S,\,\mathbb{Q})\longrightarrow
H^{1}(\Sigma_{g},\,\mathbb{Q})$ coincides with the invariant subspace
$H^{1}(\Sigma_{g},\,\mathbb{Q})^{\pi_{1}(\Sigma_{b})}$; thus, given any
automorphism $\xi\colon H^{1}(\Sigma_{g},\,\mathbb{Q})\longrightarrow
H^{1}(\Sigma_{g},\,\mathbb{Q})$ in the image of $\rho_{\pi_{1}(\Sigma_{b})}$,
we get $\xi\circ\iota^{*}=\iota^{*}$. Using (127), this in turn implies

(128) $\xi\circ h^{*}=\xi\circ(\iota^{*}\circ
f^{*}\circ\pi_{i}^{*})=(\xi\circ\iota^{*})\circ
f^{*}\circ\pi_{i}^{*}=\iota^{*}\circ f^{*}\circ\pi_{i}^{*}=h^{*},$

so $h^{*}$ is $\rho_{\pi_{1}(\Sigma_{b})}$-invariant and we are done. ∎

######  Corollary 4.17.

Let $f\colon S\longrightarrow\Sigma_{b}\times\Sigma_{b}$ be a diagonal double
Kodaira fibration as above. Then the following are equivalent.

  * $\boldsymbol{(1)}$

The pull-back map $f^{*}\colon
H^{1}(\Sigma_{b}\times\Sigma_{b},\,\mathbb{Q})\longrightarrow
H^{1}(S,\,\mathbb{Q})$ is an isomorphism.

  * $\boldsymbol{(2)}$

$H^{1}(\Sigma_{g},\,\mathbb{Q})^{G}=H^{1}(\Sigma_{g},\,\mathbb{Q})^{\pi_{1}(\Sigma_{b})}$.

###### Proof.

It is sufficient to show that both conditions are equivalent to
$\mathsf{b}_{1}(S)=4b$. For $\boldsymbol{(1)}$, this follows from the
injectivity of the pull-back in cohomology associated with a finite $G$-cover.
On the other hand, by Proposition 4.16, equality $\boldsymbol{(2)}$ holds if
and only if

(129) $\dim H^{1}(\Sigma_{b},\,\mathbb{Q})=\dim H^{1}(S,\,\mathbb{Q})-2b,$

namely, if and only if $\dim H^{1}(S,\,\mathbb{Q})=4b$, as claimed. ∎

Borrowing the terminology from [Breg18, Definition 2.8], we say that a
diagonal double Kodaira fibration $f\colon
S\longrightarrow\Sigma_{b}\times\Sigma_{b}$ is _maximal_ if it satisfies one
of the equivalent conditions in Corollary 4.17. Since
$\mathsf{b}_{1}(\Sigma_{2}\times\Sigma_{2})=8$, Proposition 4.14 implies the
following

######  Proposition 4.18.

Let $f\colon S\longrightarrow\Sigma_{2}\times\Sigma_{2}$ be a diagonal double
Kodaira fibration, associated with a diagonal double Kodaira structure of type
$(b,\,n)=(2,\,2)$ on an extra-special group $G$ of order $32$. Then $f$ is
maximal.

## Acknowledgments

F. Polizzi was partially supported by GNSAGA-INdAM. He thanks Andrea Causin
for drawing the figures and Zönke Rollenske for answering some of his
questions via e-mail. Both authors are grateful to Ian Agol, Yves de
Cornulier, “Jonathan”, Derek Holt, Max Horn, Moishe Kohan, Roberto Pignatelli,
“Primoz”, Geoff Robinson, John Shareshian, Remy van Dobben de Bruyn, Will
Sawin for their precious answers and comments in the MathOverflow threads

https://mathoverflow.net/questions/357453
https://mathoverflow.net/questions/366044
https://mathoverflow.net/questions/366771
https://mathoverflow.net/questions/368628
https://mathoverflow.net/questions/371181
https://mathoverflow.net/questions/379272
https://mathoverflow.net/questions/380292
https://mathoverflow.net/questions/390447

## Appendix A. Non abelian groups of order $24$ and $32$

$\mathrm{IdSmallGroup}(G)$ | $G$ | $\mathrm{Presentation}$
---|---|---
$G(24,\,1)$ | $\mathsf{D}_{8,\,3,\,-1}$ | $\langle x,\,y\;|\;x^{8}=y^{3}=1,\,xyx^{-1}=y^{-1}\rangle$
$G(24,\,3)$ | $\mathsf{SL}(2,\,\mathbb{F}_{3})$ | $\langle x,\,y,\,z\;|\;\,x^{3}=y^{3}=z^{2}=xyz\rangle$
$G(24,\,4)$ | $\mathsf{Q}_{24}$ | $\langle x,\,y,\,z\;|\;\,x^{6}=y^{2}=z^{2}=xyz\rangle$
$G(24,\,5)$ | $\mathsf{D}_{2,\,12,\,5}$ | $\langle x,\,y\;|\;x^{2}=y^{12}=1,\,xyx^{-1}=y^{5}\rangle$
$G(24,\,6)$ | $\mathsf{D}_{24}$ | $\langle x,\,y\;|\;x^{2}=y^{12}=1,\,xyx^{-1}=y^{-1}\rangle$
$G(24,\,7)$ | $\mathbb{Z}_{2}\times\mathsf{D}_{4,\,3,\,-1}$ | $\langle z\;|\;z^{2}=1\rangle\times\langle x,\,y\;|\;x^{4}=y^{3}=1,\,xyx^{-1}=y^{-1}\rangle$
$G(24,\,8)$ | $((\mathbb{Z}_{2})^{2}\times\mathbb{Z}_{3})\rtimes\mathbb{Z}_{2}$ | $\langle x,\,\,y,\,z,\,w\;|\;x^{2}=y^{2}=z^{2}=w^{3}=1,$
|  | $[y,z]=[y,w]=[z,w]=1,$
|  | $xyx^{-1}=y,\;xzx^{-1}=zy,\;xwx^{-1}=w^{-1}\rangle$
$G(24,\,10)$ | $\mathbb{Z}_{3}\times\mathsf{D}_{8}$ | $\langle z\;|\;z^{3}=1\rangle\times\langle x,\,y\;|\;x^{2}=y^{4}=1,\,xyx^{-1}=y^{-1}\rangle$
$G(24,\,11)$ | $\mathbb{Z}_{3}\times\mathsf{Q}_{8}$ | $\langle z\;|\;z^{3}=1\rangle\times\langle i,\,j,\,k\;|\;i^{2}=j^{2}=k^{2}=ijk\rangle$
$G(24,\,12)$ | $\mathsf{S}_{4}$ | $\langle x,\,y\;|\;x=(12),\,y=(1234)\rangle$
$G(24,\,13)$ | $\mathbb{Z}_{2}\times\mathsf{A}_{4}$ | $\langle z\;|\;z^{2}=1\rangle\times\langle x,\,y\;|\;x=(12)(34),y=(123)\rangle$
$G(24,\,14)$ | $(\mathbb{Z}_{2})^{2}\times\mathsf{S}_{3}$ | $\langle z,\,w\;|\;z^{2}=w^{2}=[z,\,w]=1\rangle$
|  | $\times\langle x,\,y\;|\;x=(12),\,y=(123)\rangle$
Table 1. Nonabelian groups of order $24$.
Source: groupprops.subwiki.org/wiki/Groups_of_order_24 $\mathrm{IdSmallGroup}(G)$ | $G$ | $\mathrm{Presentation}$
---|---|---
$G(32,\,2)$ | $(\mathbb{Z}_{4}\times\mathbb{Z}_{2})\rtimes\mathbb{Z}_{4}$ | $\langle x,\,y,\,z\;|\;x^{4}=y^{4}=z^{2}=1,$
|  | $[x,\,y]=z,\,[x,\,z]=[y,\,z]=1\rangle$
$G(32,\,4)$ | $\mathsf{D}_{4,\,8,\,5}$ | $\langle x,\,y\;|\;x^{4}=y^{8}=1,xyx^{-1}=y^{5}\rangle$
$G(32,\,5)$ | $(\mathbb{Z}_{8}\times\mathbb{Z}_{2})\rtimes\mathbb{Z}_{2}$ | $\langle x,\,y,\,z\;|\;x^{8}=y^{2}=z^{2}=1,\,[x,\,y]=1,$
|  | $zxz^{-1}=x^{5}y,\,zyz^{-1}=y\rangle$
$G(32,\,6)$ | $(\mathbb{Z}_{2})^{3}\rtimes\mathbb{Z}_{4}$ | $\langle x,\,y,\,z,\,w\mid x^{2}=y^{2}=z^{2}=w^{4}=1,$
|  | $[x,\,y]=1,\,[x,\,z]=1,\,[y,\,z]=1,$
|  | $wxw^{-1}=x,\,wyw^{-1}=xy,\,wzw^{-1}=yz\rangle$
$G(32,\,7)$ | $(\mathbb{Z}_{8}\rtimes\mathbb{Z}_{2})\rtimes\mathbb{Z}_{2}$ | $\langle x,\,y,\,z,\,u,\,w\mid y^{2}=z^{2}=w^{2}=1,$
|  | $u^{2}=w^{-1},\,x^{2}=u,\,(yz)^{2}=1,\,(yu^{-1})^{2}=1,$
|  | $uzu^{-1}=z^{-1},\,xyzx^{-1}=y^{-1}\rangle$
$G(32,\,8)$ | $(\mathbb{Z}_{2})^{2}\,.\,(\mathbb{Z}_{4}\times\mathbb{Z}_{2})$ | $\langle x,\,y,\,z\mid x^{8}=y^{2}=1,\,z^{2}=x^{4},$
|  | $xy=yx^{5},\,[y,\,z]=1,\,xz=zxy^{-1}\rangle$
$G(32,\,9)$ | $(\mathbb{Z}_{8}\times\mathbb{Z}_{2})\rtimes\mathbb{Z}_{2}$ | $\langle x,\,y,\,z\;|\;x^{8}=y^{2}=z^{2}=1,\,[x,\,y]=1,$
|  | $zxz^{-1}=x^{3}y,\,zyz^{-1}=y\rangle$
$G(32,\,10)$ | $\mathsf{Q}_{8}\rtimes\mathbb{Z}_{4}$ | $\langle i,\,j,\,k,\,x\mid i^{2}=j^{2}=k^{2}=ijk,\,x^{4}=1,$
|  | $xix^{-1}=j,\,xjx^{-1}=i,\,xkx^{-1}=k^{-1}\rangle$
$G(32,\,11)$ | $(\mathbb{Z}_{4})^{2}\rtimes\mathbb{Z}_{2}$ | $\langle x,\,y,\,z\mid x^{4}=y^{4}=[x,\,y]=1,\,z^{2}=1,$
|  | $zxz^{-1}=y,\,zyz^{-1}=x\rangle$
$G(32,\,12)$ | $\mathsf{D}_{8,\,4,\,3}$ | $\langle x,\,y\;|\;x^{8}=y^{4}=1,xyx^{-1}=y^{3}\rangle$
$G(32,\,13)$ | $\mathsf{D}_{4,\,8,\,3}$ | $\langle x,\,y\;|\;x^{4}=y^{8}=1,xyx^{-1}=y^{3}\rangle$
$G(32,\,14)$ | $\mathsf{D}_{4,\,8,\,-1}$ | $\langle x,\,y\;|\;x^{4}=y^{8}=1,xyx^{-1}=y^{-1}\rangle$
$G(32,\,15)$ | $\mathbb{Z}_{4}\,.\,\mathsf{D}_{8}$ | $\langle x,\,y,\,z,\,u,\,w\mid w^{2}=1,\,z^{2}=u^{2}=w^{-1},$
|  | $x^{2}=u,\,y^{2}=z,\,xzx^{-1}=z^{-1},$
|  | $[y,\,u]=1,\,xyxu=y^{-1}\rangle$
$G(32,\,17)$ | $\mathsf{D}_{2,\,16,\,9}$ | $\langle x,\,y\mid x^{2}=y^{16}=1,\,xyx^{-1}=y^{9}\rangle$
$G(32,\,18)$ | $\mathsf{D}_{32}$ | $\langle x,\,y\mid x^{2}=y^{16}=1,xyx^{-1}=y^{-1}\rangle$
$G(32,\,19)$ | $\mathsf{QD}_{32}$ | $\langle x,\,y\mid x^{2}=y^{16}=1,xyx^{-1}=y^{7}\rangle$
$G(32,\,20)$ | $\mathsf{Q}_{32}$ | $\langle x,\,y,\,z\mid x^{8}=y^{2}=z^{2}=xyz\rangle$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{G(32,\,22)}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{Z}_{2}\times((\mathbb{Z}_{4}\times\mathbb{Z}_{2})\rtimes\mathbb{Z}_{2})}$ | $\langle w\mid w^{2}=1\rangle\times$
|  | $\langle x,\,y,\,z\mid x^{4}=y^{2}=z^{2}=1,\,[x,\,y]=1,$
|  | $zxz^{-1}=xy,\,zyz^{-1}=y\rangle$
$G(32,\,23)$ | $\mathbb{Z}_{2}\times\mathsf{D}_{4,\,4,\,3}$ | $\langle z\mid z^{2}=1\rangle\times\langle x,\,y\mid x^{4}=y^{4}=1,\,xyx^{-1}=y^{3}\rangle$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{G(32,\,24)}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{(\mathbb{Z}_{4})^{2}\rtimes\mathbb{Z}_{2}}$ | $\langle x,\,y,\,z\mid x^{4}=y^{4}=z^{2}=1,$
|  | $[x,\,y]=1,\,zxz^{-1}=x,\,zyz^{-1}=x^{2}y\rangle$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{G(32,\,25)}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{Z}_{4}\times\mathsf{D}_{8}}$ | $\langle z\mid z^{4}=1\rangle\times\langle x,\,y\mid x^{2}=y^{4}=1,\,xyx^{-1}=y^{-1}\rangle$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{G(32,\,26)}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathbb{Z}_{4}\times\mathsf{Q}_{8}}$ | $\langle z\mid z^{4}=1\rangle\times\langle i,\,j,\,k\mid i^{2}=j^{2}=k^{2}=ijk\rangle$
$\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{G(32,\,27)}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{(\mathbb{Z}_{2})^{3}\rtimes(\mathbb{Z}_{2})^{2}}$ | $\langle x,\,y,\,z,\,a,\,b\mid$
|  | $x^{2}=y^{2}=z^{2}=a^{2}=b^{2}=1,$
|  | $[x,\,y]=[y,\,z]=[x,\,z]=[a,\,b]=1,$
|  | $axa^{-1}=x,\,aya^{-1}=y,\,aza^{-1}=xz,$
|  | $bxb^{-1}=x,\,byb^{-1}=y,\,bzb^{-1}=yz\rangle$
$G(32,\,28)$ | $(\mathbb{Z}_{4}\times(\mathbb{Z}_{2})^{2})\rtimes\mathbb{Z}_{2}$ | $\langle x,\,y,\,z,\,w\mid x^{4}=y^{2}=z^{2}=w^{2}=1,$
|  | $[x,y]=[x,\,z]=[y,\,z]=1,$
|  | $wxw^{-1}=x^{-1},\,wyw^{-1}=z,\,wzw^{-1}=y\rangle$
$G(32,\,29)$ | $(\mathbb{Z}_{2}\times\mathsf{Q}_{8})\rtimes\mathbb{Z}_{2}$ | $\langle x,\,i,\,j,\,k,\,z\mid x^{2}=z^{2}=1,\,i^{2}=j^{2}=k^{2}=ijk,$
|  | $[x,\,i]=[x,\,j]=[x,\,k]=1,$
|  | $zxz^{-1}=x,\,ziz^{-1}=i,\,zjz^{-1}=xj^{-1}\rangle$
$G(32,\,30)$ | $(\mathbb{Z}_{4}\times(\mathbb{Z}_{2})^{2})\rtimes\mathbb{Z}_{2}$ | $\langle x,\,y,\,z,\,w\mid x^{4}=y^{2}=z^{2}=w^{2}=1,$
|  | $[x,y]=[x,\,z]=[y,\,z]=1,$
|  | $wxw^{-1}=xy,\,wyw^{-1}=y,\,wzw^{-1}=x^{2}z\rangle$

$\mathrm{IdSmallGroup}(G)$ | $G$ | $\mathrm{Presentation}$
---|---|---
$G(32,\,31)$ | $(\mathbb{Z}_{4})^{2}\rtimes\mathbb{Z}_{2}$ | $\langle x,\,y,\,z\mid x^{4}=y^{4}=[x,\,y]=1,\,z^{2}=1,$
|  | $zxz^{-1}=xy^{2},\,zyz^{-1}=x^{2}y\rangle$
$G(32,\,32)$ | $(\mathbb{Z}_{2})^{2}\,.\,(\mathbb{Z}_{2})^{3}$ | $\langle x,\,y,\,z,\,u,\,w\mid u^{2}=w^{2}=1,$
|  | $u=z^{2},\,u=x^{-2},\,w=y^{-2},$
|  | $yxy^{-1}=x^{-1},\,[y,\,z]=1,\,xzxwz=1\rangle$
$G(32,\,33)$ | $(\mathbb{Z}_{4})^{2}\rtimes\mathbb{Z}_{2}$ | $\langle x,\,y,\,z\mid x^{4}=y^{4}=[x,\,y]=1,\,z^{2}=1,$
|  | $zxz^{-1}=xy^{2},\,zyz^{-1}=x^{2}y^{-1}\rangle$
$G(32,\,34)$ | $(\mathbb{Z}_{4})^{2}\rtimes\mathbb{Z}_{2}$ | $\langle x,\,y,\,z\mid x^{4}=y^{4}=[x,\,y]=1,\,z^{2}=1,$
|  | $zxz^{-1}=x^{-1},\,zyz^{-1}=y^{-1}\rangle$
$G(32,\,35)$ | $\mathbb{Z}_{4}\rtimes\mathsf{Q}_{8}$ | $\langle x,\,i,\,j,\,k\mid x^{4}=1,\,i^{2}=j^{2}=k^{2}=ijk,$
|  | $ixi^{-1}=x^{-1},\,jxj^{-1}=x^{-1},\,kxk^{-1}=x\rangle$
$G(32,\,37)$ | $(\mathbb{Z}_{8}\times\mathbb{Z}_{2})\rtimes\mathbb{Z}_{2}$ | $\langle x,\,y,\,z\;|\;x^{8}=y^{2}=z^{2}=1,\,[x,\,y]=1,$
|  | $zxz^{-1}=x^{5},\,zyz^{-1}=y\rangle$
$G(32,\,38)$ | $(\mathbb{Z}_{8}\times\mathbb{Z}_{2})\rtimes\mathbb{Z}_{2}$ | $\langle x,\,y,\,z\;|\;x^{8}=y^{2}=z^{2}=1,\,[x,\,y]=1,$
|  | $zxz^{-1}=x,\,zyz^{-1}=x^{4}y\rangle$
$G(32,\,39)$ | $\mathbb{Z}_{2}\times\mathsf{D}_{16}$ | $\langle z\mid z^{2}=1\rangle\times\langle x,\,y\mid x^{2}=y^{8}=1,xyx^{-1}=y^{-1}\rangle$
$G(32,\,40)$ | $\mathbb{Z}_{2}\times\mathsf{QD}_{16}$ | $\langle z\mid z^{2}=1\rangle\times\langle x,\,y\mid x^{2}=y^{8}=1,xyx^{-1}=y^{3}\rangle$
$G(32,\,41)$ | $\mathbb{Z}_{2}\times\mathsf{Q}_{16}$ | $\langle w\mid w^{2}=1\rangle\times\langle x,\,y,\,z\mid x^{4}=y^{2}=z^{2}=xyz\rangle$
$G(32,\,42)$ | $(\mathbb{Z}_{8}\times\mathbb{Z}_{2})\rtimes\mathbb{Z}_{2}$ | $\langle x,\,y,\,z\;|\;x^{8}=y^{2}=z^{2}=1,\,[x,\,y]=1,$
|  | $zxz^{-1}=x^{3},\,zyz^{-1}=x^{4}y\rangle$
$G(32,\,43)$ | $\mathbb{Z}_{8}\rtimes(\mathbb{Z}_{2})^{2}$ | $\langle x,\,y,\,z\mid x^{8}=1,\,y^{2}=z^{2}=[y,\,z]=1,$
|  | $yxy^{-1}=x^{-1},\,zxz^{-1}=x^{5}\rangle$
$G(32,\,44)$ | $(\mathbb{Z}_{2}\times\mathsf{Q}_{8})\rtimes\mathbb{Z}_{2}$ | $\langle x,\,i,\,j,\,k,\,z\mid x^{2}=z^{2}=1,\,i^{2}=j^{2}=k^{2}=ijk,$
|  | $[x,\,i]=[x,\,j]=[x,\,k]=1,$
|  | $zxz^{-1}=xi^{2},\,ziz^{-1}=j,\,zjz^{-1}=i\rangle$
$G(32,\,46)$ | $(\mathbb{Z}_{2})^{2}\times\mathsf{D}_{8}$ | $\langle z,\,w\mid z^{2}=w^{2}=[z,\,w]=1\rangle$
|  | $\times\langle x,\,y\mid x^{2}=y^{4}=1,\,xyx^{-1}=y^{-1}\rangle$
$G(32,\,47)$ | $(\mathbb{Z}_{2})^{2}\times\mathsf{Q}_{8}$ | $\langle z,\,w\mid z^{2}=w^{2}=[z,\,w]=1\rangle$
|  | $\times\langle i,\,j,\,k\mid i^{2}=j^{2}=k^{2}=ijk\rangle$
$G(32,\,48)$ | $(\mathbb{Z}_{4}\times(\mathbb{Z}_{2})^{2})\rtimes\mathbb{Z}_{2}$ | $\langle x,\,y,\,z,\,w\mid x^{4}=y^{2}=z^{2}=w^{2}=1,$
|  | $[x,y]=[x,\,z]=[y,\,z]=1,$
|  | $wxw^{-1}=x,\,wyw^{-1}=y,\,wzw^{-1}=x^{2}z\rangle$
$G(32,\,49)$ | $\mathsf{H}_{5}(\mathbb{Z}_{2})$ | $\langle\mathsf{r}_{1},\,\mathsf{t}_{1},\,\mathsf{r}_{2},\,\mathsf{t}_{2},\,\mathsf{z}\mid\mathsf{r}_{j}^{2}=\mathsf{t}_{j}^{2}=\mathsf{z}^{2}=1,$
|  | $[\mathsf{r}_{j},\,\mathsf{z}]=[\mathsf{t}_{j},\,\mathsf{z}]=1,$
|  | $[\mathsf{r}_{j},\,\mathsf{r}_{k}]=[\mathsf{t}_{j},\,\mathsf{t}_{k}]=1,$
|  |  $[\mathsf{r}_{j},\,\mathsf{t}_{k}]=\mathsf{z}^{-\delta_{jk}}\,\rangle,$ see (12)
$G(32,\,50)$ | $\mathsf{G}_{5}(\mathbb{Z}_{2})$ | $\langle\,\mathsf{r}_{1},\,\mathsf{t}_{1},\,\mathsf{r}_{2},\,\mathsf{t}_{2},\,\mathsf{z}\mid,\mathsf{r}_{1}^{2}=\mathsf{t}_{1}^{2}=\mathsf{z}^{2}=1,$
|  | $\mathsf{r}_{2}^{2}=\mathsf{t}_{2}^{2}=\mathsf{z},$
|  | $[\mathsf{r}_{j},\,\mathsf{z}]=[\mathsf{t}_{j},\,\mathsf{z}]=1,$
|  | $[\mathsf{r}_{j},\,\mathsf{r}_{k}]=[\mathsf{t}_{j},\,\mathsf{t}_{k}]=1,$
|  |  $[\mathsf{r}_{j},\,\mathsf{t}_{k}]=\mathsf{z}^{-\delta_{jk}}\,\rangle,$ see (13)
Table 2. Nonabelian groups of order $32$.
Source: groupprops.subwiki.org/wiki/Groups_of_order_32