# Birational classification for algebraic tori

Akinari Hoshi  Department of Mathematics, Niigata University, Niigata
950-2181, Japan<EMAIL_ADDRESS>and  Aiichi Yamasaki
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
<EMAIL_ADDRESS>
###### Abstract.

We give a stably birational classification for algebraic $k$-tori of
dimensions $3$ and $4$ over a field $k$. Kunyavskii [Kun90] proved that there
exist $15$ not stably $k$-rational cases among $73$ cases of algebraic
$k$-tori of dimension $3$. Hoshi and Yamasaki [HY17] showed that there exist
exactly $487$ (resp. $7$, resp. $216$) stably $k$-rational (resp. not stably
but retract $k$-rational, resp. not retract $k$-rational) cases of algebraic
$k$-tori of dimension $4$. First, we define the weak stably $k$-equivalence of
algebraic $k$-tori and show that there exist $13$ (resp. $128$) weak stably
$k$-equivalent classes of algebraic $k$-tori $T$ of dimension $3$ (resp. $4$)
which are not stably $k$-rational by computing some cohomological stably
birational invariants, e.g. the Brauer-Grothendieck group of $X$ where $X$ is
a smooth $k$-compactification of $T$, provided by Kunyavskii, Skorobogatov and
Tsfasman [KST89]. We make a procedure to compute such stably birational
invariants effectively and the computations are done by using the computer
algebra system GAP. Second, we define the $p$-part of the flabby class
$[\widehat{T}]^{fl}$ as a $\mathbbm{Z}_{p}[Sy_{p}(G)]$-lattice and prove that
they are faithful and indecomposable $\mathbbm{Z}_{p}[Sy_{p}(G)]$-lattices
unless it vanishes for $p=2$ (resp. $p=2,3)$ in dimension $3$ (resp. $4$). The
$\mathbbm{Z}_{p}$-ranks of them are also given. Third, we give a necessary and
sufficient condition for which two not stably $k$-rational algebraic $k$-tori
$T$ and $T^{\prime}$ of dimensions $3$ (resp. $4$) are stably birationally
$k$-equivalent in terms of the splitting fields and the weak stably
$k$-equivalent classes of $T$ and $T^{\prime}$. In particular, the splitting
fields of them should coincide if $\widehat{T}$ and $\widehat{T}^{\prime}$ are
indecomposable. Forth, for each $7$ cases of not stably but retract
$k$-rational algebraic $k$-tori of dimension $4$, we find an algebraic
$k$-torus $T^{\prime}$ of dimension $4$ which satisfies that
$T\times_{k}T^{\prime}$ is stably $k$-rational. Finally, we give a criteria to
determine whether two algebraic $k$-tori $T$ and $T^{\prime}$ of general
dimensions are stably birationally $k$-equivalent when $T$ (resp.
$T^{\prime}$) is stably birationally $k$-equivalent to some algebraic
$k$-torus $T^{\prime\prime}$ of dimension up to $4$.

###### 2010 Mathematics Subject Classification:

Primary 11E72, 12F20, 13A50, 14E08, 20C10, 20G15.

Key words and phrases. Rationality problem, birational classification,
algebraic tori, flabby resolution, Krull-Schmidt-Azumaya theorem.
This work was partially supported by KAKENHI (24540019, 25400027, 16K05059,
19K03418, 20K03511).

###### Contents

  1. 1 Introduction
  2. 2 The groups $N_{3,i}$, $N_{31,i}$, $N_{4,i}$ and $I_{4,i}$: not stably $k$-rational cases of dimensions $3$ and $4$
  3. 3 The subgroups $N_{3,j}\leq N_{3,i}$, $N_{31,j}\leq N_{31,i}$, $N_{31,l},N_{4,j},I_{4,m}\leq N_{4,i}$ and $I_{4,j}\leq I_{4,i}$
  4. 4 Stably birational invariants: torus invariants
  5. 5 Weak stably $k$-equivalent classes via torus invariants
  6. 6 Proofs of Theorem 1.22 and Theorem 1.24: weak stably $k$-equivalent classes
  7. 7 The $p$-Part of the flabby class $[M_{G}]^{fl}$ as a $\mathbbm{Z}_{p}[Sy_{p}(G)]$-lattice
  8. 8 Proofs of ${\rm(1)}\Rightarrow{\rm(2)}$ of Theorem 1.23, Theorem 1.26 and Theorem 1.29: stably $k$-equivalent classes
  9. 9 Proofs of ${\rm(2)}\Rightarrow{\rm(1)}$ of Theorem 1.23, Theorem 1.26 and Theorem 1.29: stably $k$-equivalent classes
  10. 10 Proofs of ${\rm(2)}\Leftrightarrow{\rm(3)}\Leftrightarrow{\rm(4)}$ of Theorem 1.23, Theorem 1.26 and Theorem 1.29
  11. 11 Proof of Theorem 1.31: relationships among the $I_{4,i}$ cases
  12. 12 Proof of Theorem 1.32: higher dimensions

##  1\. Introduction

We first recall some relevant definitions of rationality of function fields.

######  Definition 1.1.

Let $K/k$ and $K^{\prime}/k$ be finitely generated field extensions of a base
field $k$.
(1) $K$ is called rational over $k$ (or $k$-rational for short) if $K$ is
purely transcendental over $k$, i.e. $K$ is isomorphic to
$k(x_{1},\ldots,x_{n})$, the rational function field over $k$ with $n$
variables $x_{1},\ldots,x_{n}$ for some integer $n$;
(2) $K$ is called stably $k$-rational if $K(y_{1},\ldots,y_{m})$ is
$k$-rational for some algebraically independent elements $y_{1},\ldots,y_{m}$
over $K$;
(3) $K$ and $K^{\prime}$ are called stably $k$-equivalent (or stably
$k$-isomorphic) if $K(y_{1},\ldots,y_{m})\simeq
K^{\prime}(z_{1},\ldots,z_{n})$ for some algebraically independent elements
$y_{1},\ldots,y_{m}$ over $K$ and $z_{1},\ldots,z_{n}$ over $K^{\prime}$;
(4) When $k$ is an infinite field, $K$ is called retract $k$-rational if there
exists a $k$-algebra domain $R\subset K$ such that (i) $K$ is the quotient
field of $R$ and (ii) the identity map $1_{R}:R\rightarrow R$ factors through
a localized polynomial ring over $k$, i.e. there exist a non-zero polynomial
$f\in k[x_{1},\ldots,x_{n}]$ and $k$-algebra homomorphisms
$\varphi:R\rightarrow k[x_{1},\ldots,x_{n}][1/f]$ and
$\psi:k[x_{1},\ldots,x_{n}][1/f]\rightarrow R$ satisfying
$\psi\circ\varphi=1_{R}$ (cf. Saltman [Sal84, Definition 3.1]);
(5) $K$ is called $k$-unirational if $k\subset K\subset k(x_{1},\ldots,x_{n})$
for some integer $n$.

It is not difficult to see that

“$k$-rational” $\Rightarrow$ “stably $k$-rational” $\Rightarrow$ “retract
$k$-rational” $\Rightarrow$ “$k$-unirational”.

Let $\overline{k}$ be a fixed separable closure of the base field $k$. Let $T$
be an algebraic $k$-torus, i.e. a group $k$-scheme with fiber product (base
change) $T\times_{k}\overline{k}=T\times_{{\rm Spec}\,k}\,{\rm
Spec}\,\overline{k}\simeq(\mathbbm{G}_{m,\overline{k}})^{n}$; $k$-form of the
split torus $(\mathbbm{G}_{m})^{n}$.

######  Definition 1.2.

An algebraic $k$-torus $T$ is said to be $k$-rational (resp. stably
$k$-rational, retract $k$-rational, $k$-unirational) if the function field
$k(T)$ of $T$ over $k$ is $k$-rational (resp. stably $k$-rational, retract
$k$-rational, $k$-unirational). Two algebraic $k$-tori $T$ and $T^{\prime}$
are said to be birationally $k$-equivalent $(k$-isomorphic$)$ (resp. stably
birationally $k$-equivalent), denoted by $T\approx T^{\prime}$ (resp.
$T\stackrel{{\scriptstyle\rm s.b.}}{{\approx}}T^{\prime}$), if their function
fields $k(T)$ and $k(T^{\prime})$ are $k$-isomorphic (resp. stably
$k$-equivalent). For an equivalent definition in the language of algebraic
geometry, see e.g. Colliot-Thélène and Sansuc [CTS07, Section 1], Beauville
[Bea16].

We consider the stably birational classification for algebraic $k$-tori.

######  Problem 1.3 (Stably birational classification for algebraic $k$-tori).

For given two algebraic $k$-tori $T$ and $T^{\prime}$, determine whether $T$
and $T^{\prime}$ are stably birationally $k$-equivalent.

Let $G$ be a finite group and $M$ be a $\mathbbm{Z}[G]$-lattice (or just
$G$-lattice for short), i.e. finitely generated $\mathbbm{Z}[G]$-module which
is $\mathbbm{Z}$-free as an abelian group. There exists the minimal finite
Galois extension $L/k$ with Galois group $G={\rm Gal}(L/k)$ such that $T$
splits over $L$: $T\times_{k}L\simeq(\mathbbm{G}_{m,L})^{n}$. The category of
$G$-lattices is anti-equivalent to the category of algebraic $k$-tori which
split over $L$ (see Ono [Ono61, Section 1.2], Voskresenskii [Vos98, page 27,
Example 6], Knus, Merkurjev, Rost and Tignol [KMRT98, Proposition 20.17]).
Indeed, if $T$ is an algebraic $k$-torus, then the character module
$\widehat{T}={\rm Hom}(T,\mathbbm{G}_{m})$ of $T$ becomes a $G$-lattice.
Conversely, for a $G$-lattice $M$, there exists an algebraic $k$-torus $T={\rm
Spec}(L[M]^{G})$ which splits over $L$ such that $\widehat{T}\simeq M$ as
$G$-lattices. The invariant field $L(M)^{G}$ may be identified with the
function field $k(T)$ of $T$ and hence it is $k$-unirational (see Endo and
Miyata [EM74], [Vos98, page 40, Example 21]). Isomorphism classes of $k$-tori
of dimension $n$ correspond bijectively to the elements of the set
$H^{1}(\mathcal{G},{\rm GL}(n,\mathbbm{Z}))$ via Galois descent where
$\mathcal{G}={\rm Gal}(\overline{k}/k)$ since ${\rm
Aut}(\mathbbm{G}_{m}^{n})={\rm GL}(n,\mathbbm{Z})$. The algebraic $k$-torus
$T$ of dimension $n$ is uniquely determined by the integral representation
$h:\mathcal{G}\rightarrow{\rm GL}(n,\mathbbm{Z})$ up to conjugacy, and
$G=h(\mathcal{G})$ is a finite subgroup of ${\rm GL}(n,\mathbbm{Z})$ (see
[Vos98, page 57, Section 4.9]). The minimal splitting field $L$ of $T$
corresponds to $\mathcal{H}$ where $\rho:\mathcal{G}\rightarrow G$ and
$\mathcal{H}={\rm Ker}(\rho)$ with $G\simeq\mathcal{H}/\mathcal{G}$.

We recall some basic tools on lattice arguments in order to study the (stably,
retract) rationality problem of algebraic $k$-tori (see Manin [Man74], [Man86,
Appendix], Colliot-Thélène and Sansuc [CTS77, Section 1], Swan [Swa83, Section
8], [Swa10], Manin and Tsfasman [MT86, Section 4], Kunyavskii, Skorobogatov,
Tsfasman [KST89, Section 2], Voskresenskii [Vos98, Chapter 2], Cortella and
Kunyavskii [CK00], Lorenz [Lor05, Chapter 2], Colliot-Thélène [CT07, Section
5]).

######  Definition 1.4.

Let $\mathcal{G}$ be a profinite group and $M$ be a (continuous)
$\mathcal{G}$-lattice.
(1) $M$ is called permutation $\mathcal{G}$-lattice if $M$ has a
$\mathbbm{Z}$-basis permuted by $\mathcal{G}$, i.e. $M\simeq\oplus_{1\leq
i\leq m}\mathbbm{Z}[\mathcal{G}/\mathcal{H}_{i}]$ for some open subgroups
$\mathcal{H}_{1},\ldots,\mathcal{H}_{m}$ of $\mathcal{G}$.
(2) $M$ is called stably permutation $\mathcal{G}$-lattice if $M\oplus P\simeq
P^{\prime}$ for some permutation $\mathcal{G}$-lattices $P$ and $P^{\prime}$.
(3) $M$ is called invertible if it is a direct summand of a permutation
$\mathcal{G}$-lattice, i.e. $P\simeq M\oplus M^{\prime}$ for some permutation
$\mathcal{G}$-lattice $P$ and a $\mathcal{G}$-lattice $M^{\prime}$.
(4) $M$ is called coflabby if $H^{1}(\mathcal{H},M)=0$ for any closed subgroup
$\mathcal{H}$ of $\mathcal{G}$.
(5) $M$ is called flabby if $\widehat{H}^{-1}(\mathcal{H},M)=0$ for any closed
subgroup $\mathcal{H}$ of $\mathcal{G}$ where $\widehat{H}$ is the Tate
cohomology.

It is not difficult to see

$\displaystyle\textrm{``}M\textrm{\ is\
permutation''}\Rightarrow\textrm{``}M\textrm{\ is\ stably\
permutation''}\Rightarrow\textrm{``}M\textrm{\ is\
invertible''}\Rightarrow\textrm{``}M\textrm{\ is\ flabby\ and\ coflabby''}.$

Note that the above implications in each step cannot be reversed (see e.g.
Hoshi and Yamasaki [HY17, Chapter 1]).

Let $X$ be a smooth $k$-compactification of $T$, i.e. smooth projective
$k$-variety $X$ containing $T$ as a dense open subvariety, and
$\overline{X}=X\times_{k}\overline{k}$ where $\overline{k}$ is a fixed
separable closure of the base field $k$. There exists such a smooth
$k$-compactification of an algebraic $k$-torus $T$ over any field $k$ (due to
Hironaka [Hir64] for ${\rm char}\,k=0$, see Colliot-Thélène, Harari and
Skorobogatov [CTHS05, Corollaire 1] for any field $k$).

######  Theorem 1.5 (Voskresenskii [Vos69, Section 4, page 1213], [Vos70,
Section 3, page 7], see also [Vos74], [Vos98, Section 4.6], Kunyavskii [Kun07,
Theorem 1.9] and Colliot-Thélène [CT07, Theorem 5.1, page 19] for any field
$k$).

Let $k$ be a field and $\mathcal{G}={\rm Gal}(\overline{k}/k)$. Let $T$ be an
algebraic $k$-torus, $X$ be a smooth $k$-compactification of $T$ and
$\overline{X}=X\times_{k}\overline{k}$. Then there exists an exact sequence of
$\mathcal{G}$-lattices

$\displaystyle 0\to\widehat{T}\to\widehat{Q}\to{\rm Pic}\,\overline{X}\to 0$

where $\widehat{Q}$ is permutation and ${\rm Pic}\ \overline{X}$ is flabby.

######  Theorem 1.6 (Voskresenskii [Vos70, Theorem 1], [Vos73, Theorem 3],
[Vos98, Chapter 2]).

Let $k$ be a field and $\mathcal{G}={\rm Gal}(\overline{k}/k)$. Let $T$ be an
algebraic $k$-torus, $X$ be a smooth $k$-compactification of $T$ and
$\overline{X}=X\times_{k}\overline{k}$.
(i) $T$ is stably $k$-rational if and only if $[{\rm Pic}\,\overline{X}]=0$ as
a $\mathcal{G}$-lattice.
(ii) $T$ and $T^{\prime}$ are stably birationally $k$-equivalent if and only
if $[{\rm Pic}\,\overline{X}]=[{\rm Pic}\,\overline{X^{\prime}}]$ as
$\mathcal{G}$-lattices.

######  Definition 1.7 (see Endo and Miyata [EM74, Section 1], Voskresenskii
[Vos98, Section 4.7]).

Let $\mathcal{G}$ be a profinite group. Let $\mathcal{C}(\mathcal{G})$ be the
category of all $\mathcal{G}$-lattices. Let $\mathcal{S}(\mathcal{G})$ be the
full subcategory of $\mathcal{C}(\mathcal{G})$ of all permutation
$\mathcal{G}$-lattices and $\mathcal{D}(\mathcal{G})$ be the full subcategory
of $\mathcal{C}(\mathcal{G})$ of all invertible $\mathcal{G}$-lattices. Let

$\displaystyle\mathcal{H}^{i}(\mathcal{G})=\\{M\in\mathcal{C}(\mathcal{G})\mid\widehat{H}^{i}(\mathcal{H},M)=0\
{\rm for\ any\ closed\ subgroup}\ \mathcal{H}\leq\mathcal{G}\\}\ (i=\pm 1)$

be the class of “$\widehat{H}^{i}$-vanish” $\mathcal{G}$-lattices where
$\widehat{H}^{i}$ is the Tate cohomology. Then one has the inclusions
$\mathcal{S}(\mathcal{G})\subset\mathcal{D}(\mathcal{G})\subset\mathcal{H}^{i}(\mathcal{G})\subset\mathcal{C}(\mathcal{G})$
$(i=\pm 1)$.

######  Definition 1.8.

Let $\mathcal{G}$ be a profinite group. We say that two $\mathcal{G}$-lattices
$M_{1}$ and $M_{2}$ are similar if there exist permutation
$\mathcal{G}$-lattices $P_{1}$ and $P_{2}$ such that $M_{1}\oplus P_{1}\simeq
M_{2}\oplus P_{2}$. We denote the set of similarity classes
$\mathcal{C}(\mathcal{G})/\mathcal{S}(\mathcal{G})$ by
$\mathcal{T}(\mathcal{G})$ and the similarity class of $M$ by $[M]$.
$\mathcal{T}(\mathcal{G})$ becomes a commutative monoid with respect to the
sum $[M_{1}]+[M_{2}]:=[M_{1}\oplus M_{2}]$ and the zero $0=[P]$ where
$P\in\mathcal{S}(\mathcal{G})$.

######  Definition 1.9 (see Endo and Miyata [EM74, Lemma 1.1], Colliot-Thélène
and Sansuc [CTS77, Lemme 3], Manin [Man86, Appendix, page 286]).

Let $\mathcal{G}$ be a profinite group. For a $\mathcal{G}$-lattice $M$, there
exists a short exact sequence of $\mathcal{G}$-lattices $0\rightarrow
M\rightarrow P\rightarrow F\rightarrow 0$ where $P$ is permutation and $F$ is
flabby which is called a flabby resolution of $M$. The similarity class
$[F]\in\mathcal{T}(\mathcal{G})$ of $F$ is uniquely determined and is called
the flabby class of $M$. We denote the flabby class $[F]$ of $M$ by
$[M]^{fl}$. We say that $[M]^{fl}$ is invertible if $[M]^{fl}=[E]$ for some
invertible $\mathcal{G}$-lattice $E$.

Note that (i) if $M$ is (stably) permutation (resp. invertible), then
$[M]^{fl}=0$ (resp. $[M]^{fl}$ is invertible) and (ii) if $[M]^{fl}=0$ (resp.
$[M]^{fl}$ is invertible) as a $\mathcal{G}$-lattice, then $[M]^{fl}=0$ (resp.
$[M]^{fl}$ is invertible) as an $\mathcal{H}$-lattice for any closed subgroup
$\mathcal{H}\leq\mathcal{G}$ (see Hoshi and Yamasaki [HY17, Lemma 2.17]).

From $\mathcal{G}$-lattice to $G$-lattice.

Let $L$ be the minimal splitting field of $T$ with Galois group $G={\rm
Gal}(L/k)\simeq\mathcal{G}/\mathcal{H}$. By Theorem 1.5, we obtain a flabby
resolution of $\widehat{T}$:

$\displaystyle 0\to\widehat{T}\to\widehat{Q}\to{\rm Pic}\,X_{L}\to 0$

as $G$-lattices with $[\widehat{T}]^{fl}=[{\rm Pic}\ X_{L}]$ where
$\widehat{Q}$ is permutation and ${\rm Pic}\ X_{L}$ is flabby (see also
[Vos74, Section 1]). By the inflation-restriction exact sequence $0\to
H^{1}(G,{\rm Pic}\,X_{L})\xrightarrow{\rm inf}H^{1}(k,{\rm
Pic}\,\overline{X})\xrightarrow{\rm res}H^{1}(L,{\rm Pic}\,\overline{X})$, we
get ${\rm inf}:H^{1}(G,{\rm Pic}\,X_{L})\xrightarrow{\sim}H^{1}(k,{\rm
Pic}\,\overline{X})$ because $H^{1}(L,{\rm Pic}\,\overline{X})=0$.

It follows from Theorem 1.6 (ii) that

${\rm(ii)}^{\prime}$ $T$ and $T^{\prime}$ are stably birationally
$k$-equivalent if and only if $[{\rm Pic}\,X_{\widetilde{L}}]=[{\rm
Pic}\,X^{\prime}_{\widetilde{L}}]$ as $\widetilde{H}$-lattices where
$\widetilde{L}=LL^{\prime}$, $L$ $($resp. $L^{\prime}$$)$ is the minimal
splitting field of $T$ $($resp. $T^{\prime}$$)$ and $\widetilde{H}={\rm
Gal}(\widetilde{L}/k)$ (cf. [Vos74, Proposition 1]).

The group $\widetilde{H}$ becomes a subdirect product of $G={\rm Gal}(L/k)$
and $G^{\prime}={\rm Gal}(L^{\prime}/k)$, i.e. a subgroup $\widetilde{H}$ of
$G\times G^{\prime}$ with surjections $\varphi_{1}:\widetilde{H}\rightarrow G$
and $\varphi_{2}:\widetilde{H}\rightarrow G^{\prime}$. This observation yields
a concept of “weak stably $k$-equivalence”.

######  Definition 1.10.

Let $L/k$ (resp. $L^{\prime}/k$) be a finite Galois extension of fields with
Galois group $G={\rm Gal}(L/k)$ (resp. $G^{\prime}={\rm Gal}(L^{\prime}/k))$.
Let $M$ (resp. $M^{\prime}$) be a $G$-lattice (resp. $G^{\prime}$-lattice).
(i) Two flabby classes $[M]^{fl}$ and $[M^{\prime}]^{fl}$ are said to be weak
stably $k$-equivalent, denoted by $[M]^{fl}\sim[M^{\prime}]^{fl}$, if there
exists a subdirect product $\widetilde{H}\leq G\times G^{\prime}$ of $G$ and
$G^{\prime}$ with surjections $\varphi_{1}:\widetilde{H}\rightarrow G$ and
$\varphi_{2}:\widetilde{H}\rightarrow G^{\prime}$ such that
$[M]^{fl}=[M^{\prime}]^{fl}$ as $\widetilde{H}$-lattices where $\widetilde{H}$
acts on $M$ (resp. $M^{\prime})$ through the surjection $\varphi_{1}$ (resp.
$\varphi_{2}$).
(ii) Two algebraic $k$-tori $T$ and $T^{\prime}$ said to be weak stably
birationally $k$-equivalent, denoted by $T\stackrel{{\scriptstyle\rm
s.b.}}{{\sim}}T^{\prime}$, if $[\widehat{T}]^{fl}$ and
$[\widehat{T}^{\prime}]^{fl}$ are weak stably $k$-equivalent.
(iii) A weak stably $k$-equivalence for flabby classes $[\widehat{T}]^{fl}$
(or a weak stably birationally $k$-equivalence for algebraic $k$-tori $T$)
indeed yields an equivalence relation (see Remark 1.11 (2) below) and we call
this equivalent class the weak stably $k$-equivalent class of
$[\widehat{T}]^{fl}$ (or $T$) denoted by ${\rm WSEC}_{r}$ $(r\geq 0)$ with the
stably $k$-rational class ${\rm WSEC}_{0}$.

######  Remark 1.11.

(1) $T\stackrel{{\scriptstyle\rm s.b.}}{{\approx}}T^{\prime}$ $\Rightarrow$
$T\stackrel{{\scriptstyle\rm s.b.}}{{\sim}}T^{\prime}$ (see the paragraph
before Definition 1.10).
(2) Indeed, the weak stably $k$-equivalence becomes equivalence relation: (i)
Reflexivity: $[M]^{fl}\sim[M]^{fl}$ follows if we take a subdirect product
$\widetilde{H}=\\{(g,g)\mid g\in G\\}\leq G\times G$ of $G$ and $G$ itself.
(ii) Symmetry is obvious. (iii) Transitivity: If
$[M]^{fl}\sim[M^{\prime}]^{fl}$ and
$[M^{\prime}]^{fl}\sim[M^{\prime\prime}]^{fl}$, then
$[M]^{fl}=[M^{\prime}]^{fl}$ as $\widetilde{H}$-lattices and
$[M^{\prime}]^{fl}=[M^{\prime\prime}]^{fl}$ as
$\widetilde{H}^{\prime}$-lattices where $\widetilde{H}\leq G\times G^{\prime}$
is a subdirect product of $G$ and $G^{\prime}$ with surjections
$\varphi_{1}:\widetilde{H}\rightarrow G$ and
$\varphi_{2}:\widetilde{H}\rightarrow G^{\prime}$ and
$\widetilde{H}^{\prime}\leq G^{\prime}\times G^{\prime\prime}$ is a subdirect
product of $G^{\prime}$ and $G^{\prime\prime}$ with surjections
$\varphi^{\prime}_{1}:\widetilde{H}\rightarrow G^{\prime}$ and
$\varphi^{\prime}_{2}:\widetilde{H}\rightarrow G^{\prime\prime}$. We may take
a subdirect product
$\widetilde{S}:=\\{(h,h^{\prime})\in\widetilde{H}\times\widetilde{H}^{\prime}\mid\varphi_{2}(h)=\varphi^{\prime}_{1}(h^{\prime})\\}\leq\widetilde{H}\times\widetilde{H}^{\prime}$
of $\widetilde{H}$ and $\widetilde{H}^{\prime}$ with surjections
$\theta_{1}:\widetilde{S}\rightarrow\widetilde{H},(h,h^{\prime})\mapsto h$ and
$\theta_{2}:\widetilde{S}\rightarrow\widetilde{H}^{\prime},(h,h^{\prime})\mapsto
h^{\prime}$. Then $[M]^{fl}=[M^{\prime\prime}]^{fl}$ as
$\widetilde{S}$-lattices. We consider a homomorphism
$\Phi:\widetilde{S}\rightarrow G\times
G^{\prime\prime},s\mapsto(\varphi_{1}(\theta_{1}(s)),\varphi_{2}^{\prime}(\theta_{2}(s)))$
and the image $\widetilde{H}^{\prime\prime}:={\rm Im}(\Phi)$. Then
$\widetilde{H}^{\prime\prime}\leq G\times G^{\prime\prime}$ is a subdirect
product of $G$ and $G^{\prime\prime}$ and $[M]^{fl}=[M^{\prime\prime}]^{fl}$
as $\widetilde{H}^{\prime\prime}$-lattices because
$\widetilde{H}^{\prime\prime}\simeq\widetilde{S}/{\rm Ker}(\Phi)$ (see
Colliot-Thélène and Sansuc [CTS77, Lemme 2 (i), (ix)]). This implies that
$[M]^{fl}\sim[M^{\prime\prime}]^{fl}$.

Let $L/k$ be a finite Galois extension of fields with Galois group $G={\rm
Gal}(L/k)$. Let $M=\bigoplus_{1\leq i\leq n}\mathbbm{Z}\cdot u_{i}$ be a
$G$-lattice with a $\mathbbm{Z}$-basis $\\{u_{1},\ldots,u_{n}\\}$, i.e.
finitely generated $\mathbbm{Z}[G]$-module which is $\mathbbm{Z}$-free as an
abelian group. Let $G$ act on the rational function field
$L(x_{1},\ldots,x_{n})$ over $L$ with $n$ variables $x_{1},\ldots,x_{n}$ by

(1) $\displaystyle\sigma(x_{i})=\prod_{j=1}^{n}x_{j}^{a_{i,j}},\quad 1\leq
i\leq n$

for any $\sigma\in G$, when $\sigma(u_{i})=\sum_{j=1}^{n}a_{i,j}u_{j}$,
$a_{i,j}\in\mathbbm{Z}$. The field $L(x_{1},\ldots,x_{n})$ with this action of
$G$ will be denoted by $L(M)$. The invariant field $L(M)^{G}$ of $L(M)$ under
the action of $G$ as in (1) may be identified with the function field $k(T)$
of the algebraic $k$-torus $T$ over $k$ where $M=\widehat{T}$ (see Endo and
Miyata [EM74]).

The flabby class $[M]^{fl}$ plays crucial role in the rationality problem for
$L(M)^{G}$ (resp. algebraic $k$-tori $T$) as follows (see also Theorem 1.6,
Colliot-Thélène and Sansuc [CTS77, Section 2], Voskresenskii [Vos98, Section
4.6], Kunyavskii [Kun07, Theorem 1.7], Colliot-Thélène [CT07, Theorem 5.4]):

######  Theorem 1.12.

Let $L/k$ be a finite Galois extension with Galois group $G={\rm Gal}(L/k)$.
Let $M$ and $M^{\prime}$ be $G$-lattices.
(i) $($Endo and Miyata [EM73, Theorem 1.6]$)$ $[M]^{fl}=0$ if and only if
$L(M)^{G}$ is stably $k$-rational.
(ii) $($Voskresenskii [Vos74, Theorem 2]$)$ $[M]^{fl}=[M^{\prime}]^{fl}$ if
and only if $L(M)^{G}$ and $L(M^{\prime})^{G}$ are stably $k$-isomorphic, i.e.
there exist algebraically independent elements $x_{1},\ldots,x_{m}$ over
$L(M)^{G}$ and $y_{1},\ldots,y_{n}$ over $L(M^{\prime})^{G}$ such that
$L(M)^{G}(x_{1},\ldots,x_{m})\simeq L(M^{\prime})^{G}(y_{1},\ldots,y_{n})$.
(iii) $($Saltman [Sal84, Theorem 3.14]$)$ $[M]^{fl}$ is invertible if and only
if $L(M)^{G}$ is retract $k$-rational.

######  Remark 1.13.

Colliot-Thélène and Sansuc [CTS87, Proposition 7.4, page 181] showed that the
following conditions are also equivalent for algebraic $k$-tori $T$: (i)
$[\widehat{T}]^{fl}$ is invertible; (ii) there exists an algebraic $k$-torus
$T^{\prime}$ such that $T\times_{k}T^{\prime}$ is $k$-rational; (iii) there
exists an integral $k$-variety $Y$ such that $T\times_{k}Y$ is $k$-rational;
(iv) $T$ is retract $k$-rational.

Before stating main theorems, we recall some known results about the
(stably/retract) rationality of $k$-tori of small dimensions.

For dimension $1$, all the $1$-dimensional algebraic $k$-tori $T$, i.e. the
trivial torus $\mathbbm{G}_{m}$ and the norm one torus
$R_{L/k}^{(1)}(\mathbbm{G}_{m})$ with $[L:k]=2$, are $k$-rational.

######  Theorem 1.14 (Voskresenskii [Vos67]).

All the $2$-dimensional algebraic $k$-tori $T$ are $k$-rational. In
particular, for any finite subgroup $G\leq{\rm GL}(2,\mathbbm{Z})$,
$L(x_{1},x_{2})^{G}$ is $k$-rational.

Note that $T$ is $k$-rational $\Rightarrow$ $T$ is stably $k$-rational
$\Rightarrow$ $T$ is retract $k$-rational $\Rightarrow$ $H^{1}(k,{\rm
Pic}\,\overline{X})=0$ and if $k$ is a global field, then $H^{1}(k,{\rm
Pic}\,\overline{X})=0\Rightarrow A(T)=\Sha(T)=0$ (see Voskresenskii [Vos69,
Theorem 5, page 1213], Manin [Man74, Section 30], Manin and Tsfasman [MT86]
and also Hoshi, Kanai and Yamasaki [HKY, Section 1]).

Let $G\simeq{\rm Gal}(L/k)$ be a finite group and $M\simeq M_{1}\oplus M_{2}$
be a decomposable $G$-lattice. Let $N_{i}=\\{\sigma\in G\mid\sigma(v)=v\ {\rm
for\ any}\ v\in M_{i}\\}$ be the kernel of the action of $G$ on $M_{i}$
$(i=1,2)$. Then $L(M)^{G}$ is the function field of an algebraic torus $T$
with $M=\widehat{T}$ and is $k$-isomorphic to the free composite of
$L(M_{1})^{G}$ and $L(M_{2})^{G}$ over $k$ where
$L(M_{i})^{G}=(L^{N_{i}})(M_{i}^{N_{i}})^{G/N_{i}}$ is the function field of
some torus $T_{i}$ ($i=1,2$) with $M_{i}=\widehat{T}_{i}$,
$T=T_{1}\times_{k}T_{2}$ and $M_{i}$ may be regarded as a $G/N_{i}$-lattice.
There exist $13$ non-conjugate finite subgroups of ${\rm GL}(2,\mathbbm{Z})$
and $4$ (resp. $9$) of them are decomposable (resp. indecomposable).

######  Lemma 1.15.

Let $G$ be a finite group and $M\simeq M_{1}\oplus M_{2}$ be a decomposable
$G$-lattice with the flabby class $\rho_{G}(M)=[M]^{fl}$. Let $N_{1}$ be a
normal subgroup of $G$ which acts on $M_{1}$ trivially. The $G$-lattice
$M_{1}$ may be regarded as a $G/N_{1}$-lattice with the flabby class
$\rho_{G/N_{1}}(M_{1})$ as a $G/N_{1}$-lattice. Then
(i) $\rho_{G}(M)=\rho_{G}(M_{1})+\rho_{G}(M_{2})$.
(ii) $\rho_{G}(M_{1})=0$ if and only if $\rho_{G/N_{1}}(M_{1})=0$.
(iii) $\rho_{G}(M_{1})$ is invertible if and only if $\rho_{G/N_{1}}(M_{1})$
is invertible.

###### Proof.

(i) Let $0\rightarrow M_{i}\rightarrow P_{i}\rightarrow F_{i}\rightarrow 0$ be
flabby resolutions of $M_{i}$ as $G$-lattices $(i=1,2)$. Then $0\rightarrow
M\rightarrow P_{1}\oplus P_{2}\rightarrow F_{1}\oplus F_{2}\rightarrow 0$ is a
flabby resolution of $M$. Hence $\rho_{G}(M)=[F_{1}\oplus
F_{2}]=\rho_{G}(M_{1})+\rho_{G}(M_{2})$. (ii), (iii) See [CTS77, Lemme 2] and
[Kan09, Lemma 4.1]. ∎

By Theorem 1.14 and Lemma 1.15, for decomposable $3$-dimensional $k$-tori
$T=T_{1}\times_{k}T_{2}$ with dim $T_{1}=1$ and dim $T_{2}=2$, the function
fields $k(T)\simeq L(M)^{G}$ are $k$-rational where $M=M_{1}\oplus M_{2}$ with
rank $M_{1}=1$ and rank $M_{2}=2$. There exist $73$ non-conjugate finite
subgroups of ${\rm GL}(3,\mathbbm{Z})$ and $39$ (resp. $34$) of them are
decomposable (resp. indecomposable).

Let $S_{n}$ (resp. $A_{n}$, $D_{n}$, $C_{n}$) be the symmetric (resp. the
alternating, the dihedral, the cyclic) group of degree $n$ of order $n!$
(resp. $n!/2$, $2n$, $n$). Let ${}^{t}G$ be the group of transposed matrices
of $G\leq{\rm GL}(n,\mathbbm{Z})$.

Rationality (resp. stably rationality, resp. retract rationality) problem for
$3$-dimensional algebraic $k$-tori was solved by Kunyavskii [Kun90] (see also
Lemma 7.3 and Theorem 7.12).

######  Theorem 1.16 (Kunyavskii [Kun90]).

Let $L/k$ $($resp. $L^{\prime}/k)$ be a Galois extension and $G\simeq{\rm
Gal}(L/k)$ $($resp. $G^{\prime}\simeq{\rm Gal}(L^{\prime}/k))$ be a finite
subgroup of ${\rm GL}(3,\mathbbm{Z})$ which acts on
$L(M)=L(x_{1},x_{2},x_{3})$ $($resp. $L^{\prime}(M^{\prime}))$ via
$(\ref{acts})$. Then
(1) $L(M)^{G}$ is not $k$-rational if and only if $G$ is conjugate to one of
the $15$ groups $N_{3,i}$ $(1\leq i\leq 15)$ which are given as in Table $1$
$($see also Table $6$$)$. Moreover, if $L(M)^{G}$ is not $k$-rational, then it
is not retract $k$-rational;
(2) If $G$ and $G^{\prime}$ are not isomorphic as abstract groups, then
$L(M)^{G}$ and $L^{\prime}(M^{\prime})^{G^{\prime}}$ are not stably
$k$-equivalent.

Table $1$: $L(M)^{G}$ not retract $k$-rational, rank $M=3$, $M$:
indecomposable ($15$ cases)

${}^{t}G$ in [Kun90] | $G$
---|---
$U_{1}$ | $C_{2}^{2}$
$U_{2}$ | $C_{2}^{3}$
$U_{3}$ | $D_{4}$
$U_{4}$ | $D_{4}$
$U_{5}$ | $A_{4}$
${}^{t}G$ in [Kun90] | $G$
---|---
$U_{6}$ | $D_{4}\times C_{2}$
$U_{7}$ | $A_{4}\times C_{2}$
$U_{8}$ | $S_{4}$
$U_{9}$ | $S_{4}$
$U_{10}$ | $S_{4}$
${}^{t}G$ in [Kun90] | $G$
---|---
$U_{11}$ | $S_{4}\times C_{2}$
$U_{12}$ | $S_{4}\times C_{2}$
$W_{1}$ | $C_{4}\times C_{2}$
$W_{2}$ | $C_{2}^{3}$
$W_{3}$ | $A_{4}\times C_{2}$

Stably $k$-rationality and retract $k$-rationality for algebraic $k$-tori of
dimensions $4$ and $5$ are given in Hoshi and Yamasaki [HY17, Chapter 1].
There exist $710$ non-conjugate finite subgroups of ${\rm GL}(4,\mathbbm{Z})$.
The case of dimension $4$ can be stated as follows (see [HY17, Theorem 1.12]
for the case of dimension $5$):

######  Theorem 1.17 (Hoshi and Yamasaki [HY17, Theorem 1.9]).

Let $L/k$ be a Galois extension and $G\simeq{\rm Gal}(L/k)$ be a finite
subgroup of ${\rm GL}(4,\mathbbm{Z})$ which acts on
$L(M)=L(x_{1},x_{2},x_{3},x_{4})$ via $(\ref{acts})$.
(i) $L(M)^{G}$ is stably $k$-rational if and only if $G$ is conjugate to one
of the $487$ groups which are not in Tables $7,8,9$ $($[HY17, Tables $2$, $3$,
$4$]$)$.
(ii) $L(M)^{G}$ is not stably but retract $k$-rational if and only if $G$ is
conjugate to one of the $7$ groups $I_{4,i}$ $(1\leq i\leq 7)$ which are given
as in Table $9$ $($[HY17, Table $2$]$)$.
(iii) $L(M)^{G}$ is not retract $k$-rational if and only if $G$ is conjugate
to one of the $216$ groups which consist of the $64$ groups $N_{31,i}$ $(1\leq
i\leq 64)$ in Table $7$ $($[HY17, Table $3$]$)$ and the $152$ groups $N_{4,i}$
$(1\leq i\leq 152)$ in Table $8$ $($[HY17, Table $4$]$)$.

######  Definition 1.18.

(1) The $15$ groups $G=N_{3,i}\leq{\rm GL}(3,\mathbbm{Z})$ $(1\leq i\leq 15)$
for which $L(M)^{G}$ is not retract $k$-rational are defined as in Table $6$
(see also Table $1$).
(2) The $64$ groups $G=N_{31,i}\leq{\rm GL}(4,\mathbbm{Z})$ $(1\leq i\leq 64)$
for which $L(M)^{G}$ is not retract $k$-rational where $M\simeq M_{1}\oplus
M_{2}$ with ${\rm rank}$ $M_{1}=3$ and ${\rm rank}$ $M_{2}=1$ are defined as
in Table $7$.
(3) The $152$ groups $G=N_{4,i}\leq{\rm GL}(4,\mathbbm{Z})$ $(1\leq i\leq
152)$ for which $L(M)^{G}$ is not retract $k$-rational where $M$ is
indecomposable with ${\rm rank}$ $M=4$ are defined as in Table $8$.
(4) The $7$ groups $G=I_{4,i}\leq{\rm GL}(4,\mathbbm{Z})$ $(1\leq i\leq 7)$
for which $L(M)^{G}$ is not stably but retract $k$-rational with $M$
indecomposable are defined as in Table $9$.

Tables $6,7,8,9$ will be given in Section 2.

######  Remark 1.19.

(1) We do not know the $k$-rationality of $k(T)\simeq L(M)^{G}$ in Theorem
1.17 (i) for the stably $k$-rational algebraic $k$-tori $T$ of dimension $4$
with $\widehat{T}=M$. Voskresenskii conjectured that any stably $k$-rational
algebraic $k$-torus is $k$-rational (see [Vos98, Section 6.2]).
(2) For $G=I_{4,i}$ $(1\leq i\leq 7)$ with $\widehat{T}=M_{G}$, there exists
an algebraic $k$-torus $T^{\prime}$ such that $T\times_{k}T^{\prime}$ is
stably $k$-rational (see Remark 1.13). Indeed, by
$-[M_{G}]^{fl}=[[M_{G}]^{fl}]^{fl}$ (see Swan [Swa10, Lemma 3.1] and Hoshi and
Yamasaki [HY17, Lemma 2.15]), we may find such $T^{\prime}$ with dim
$T^{\prime}={\rm rank}$ $[M_{G}]^{fl}=16,16,16,16,16,16,20$ for $1\leq i\leq
7$ respectively by using the function FlabbyResolutionLowRank($G$).actionF
(see Section 6). Hoshi and Yamasaki [HY17, Theorem 1.27] also showed that
$T\times_{k}T^{\prime}$ is stably $k$-rational for $\widehat{T}=M_{G}$ and
$\widehat{T}^{\prime}=M_{G^{\prime}}$ with $G=I_{4,1}\simeq F_{20}$ and
$G^{\prime}=I_{4,2}\simeq F_{20}$ (resp. $G=I_{4,4}\simeq S_{5}$ and
$G^{\prime}=I_{4,5}\simeq S_{5}$). We will discuss the case where $G=I_{4,i}$
later (Theorem 1.31).

Let $K/k$ be a separable field extension of degree $n$ and $L/k$ be the Galois
closure of $K/k$. Let $G={\rm Gal}(L/k)$ and $H={\rm Gal}(L/K)$. The Galois
group $G$ may be regarded as a transitive subgroup of the symmetric group
$S_{n}$ of degree $n$. Let $R^{(1)}_{K/k}(\mathbbm{G}_{m})$ be the norm one
torus of $K/k$, i.e. the kernel of the norm map
$R_{K/k}(\mathbbm{G}_{m})\rightarrow\mathbbm{G}_{m}$ where $R_{K/k}$ is the
Weil restriction (see [Vos98, page 37, Section 3.12]). The norm one torus
$R^{(1)}_{K/k}(\mathbbm{G}_{m})$ has the Chevalley module $J_{G/H}$ as its
character module and the field $L(J_{G/H})^{G}$ as its function field where
$J_{G/H}=(I_{G/H})^{\circ}={\rm Hom}_{\mathbbm{Z}}(I_{G/H},\mathbbm{Z})$ is
the dual lattice of $I_{G/H}={\rm Ker}(\varepsilon)$ and
$\varepsilon:\mathbbm{Z}[G/H]\rightarrow\mathbbm{Z}$ is the augmentation map
(see [Vos98, Section 4.8]). We have the exact sequence
$0\rightarrow\mathbbm{Z}\rightarrow\mathbbm{Z}[G/H]\rightarrow
J_{G/H}\rightarrow 0$ and rank $J_{G/H}=n-1$. Write $J_{G/H}=\oplus_{1\leq
i\leq n-1}\mathbbm{Z}x_{i}$. Then the action of $G$ on
$L(J_{G/H})=L(x_{1},\ldots,x_{n-1})$ is of the form (1).

The rationality problem for norm one tori is investigated by [EM74], [CTS77],
[Hür84], [CTS87], [LeB95], [CK00], [LL00], [Flo], [End11], [HY17], [HHY20] and
[HY21].

The aim of this paper is to give a stably birational classification for
algebraic $k$-tori of dimensions $3$ and $4$. For the case of dimension $\leq
2$, by Theorem 1.14, there exists only one algebraic $k$-torus, i.e. rational
$k$-torus, up to stably birationally $k$-equivalence.

######  Theorem 1.20 (Colliot-Thélène and Sansuc [CTS77, Corollaire, page
191]).

Let $L/k$ and $L^{\prime}/k$ be Galois extensions with ${\rm
Gal}(L/k)\simeq{\rm Gal}(L^{\prime}/k)\simeq C_{2}^{2}$. Let
$T=R_{L/k}^{(1)}(\mathbbm{G}_{m})$ and
$T^{\prime}=R_{L^{\prime}/k}^{(1)}(\mathbbm{G}_{m})$ be the corresponding norm
one tori. Then $T$ and $T^{\prime}$ are stably birationally $k$-equivalent if
and only if $L=L^{\prime}$.

The norm one tori $T=R_{L/k}^{(1)}(\mathbbm{G}_{m})$ and
$T^{\prime}=R_{L^{\prime}/k}^{(1)}(\mathbbm{G}_{m})$ as in Theorem 1.20
correspond to $U_{1}$ in Table $1$. In particular, if $k$ is a number field,
then there exist infinitely many stably birationally $k$-equivalent classes of
(non-rational) $k$-tori which correspond to $U_{1}$ in Table $1$ (see also
Voskresenskii [Vos70, Example, page 14]).

######  Definition 1.21 (The $G$-lattice $M_{G}$ of a finite subgroup $G$ of
${\rm GL}(n,\mathbbm{Z})$).

Let $G$ be a finite subgroup of ${\rm GL}(n,\mathbbm{Z})$. The $G$-lattice
$M_{G}$ of rank $n$ is defined to be the $G$-lattice with a
$\mathbbm{Z}$-basis $\\{u_{1},\ldots,u_{n}\\}$ on which $G$ acts by
$\sigma(u_{i})=\sum_{j=1}^{n}a_{i,j}u_{j}$ for any $\sigma=[a_{i,j}]\in G$
(corresponding to the equation (1)).

We state main theorems of this paper.

######  Theorem 1.22 (Main theorem 1).

Let $k$ be a field. There exist exactly $14$ weak stably birationally
$k$-equivalent classes of algebraic $k$-tori $T$ of dimension $3$ which
consist of the stably rational class ${\rm WSEC}_{0}$ and $13$ classes ${\rm
WSEC}_{r}$ $(1\leq r\leq 13)$ for $[\widehat{T}]^{fl}$ with
$\widehat{T}=M_{G}$ and $G=N_{3,i}$ $(1\leq i\leq 15)$ as in Table $2$.

Table $2$: Weak stably $k$-equivalent classes for $[\widehat{T}]^{fl}$ with
$\widehat{T}=M_{G}$ and $G=N_{3,i}$ $(1\leq i\leq 15)$

$r$ | $G=N_{3,i}:[\widehat{T}]^{fl}=[M_{G}]^{fl}\in{\rm WSEC}_{r}$ | $G$
---|---|---
$1$ | $N_{3,1}=U_{1}$ | $C_{2}^{2}$
$2$ | $N_{3,2}=U_{2}$ | $C_{2}^{3}$
$3$ | $N_{3,3}=W_{2}$ | $C_{2}^{3}$
$4$ | $N_{3,4}=W_{1}$ | $C_{4}\times C_{2}$
$5$ |  $N_{3,5}=U_{3}$, $N_{3,6}=U_{4}$ | $D_{4}$
$6$ | $N_{3,7}=U_{6}$ | $D_{4}\times C_{2}$
$7$ | $N_{3,8}=U_{5}$ | $A_{4}$
$8$ | $N_{3,9}=U_{7}$ | $A_{4}\times C_{2}$
$9$ | $N_{3,10}=W_{3}$ | $A_{4}\times C_{2}$
$10$ |  $N_{3,11}=U_{9}$, $N_{3,13}=U_{10}$ | $S_{4}$
$11$ | $N_{3,12}=U_{8}$ | $S_{4}$
$12$ | $N_{3,14}=U_{12}$ | $S_{4}\times C_{2}$
$13$ | $N_{3,15}=U_{11}$ | $S_{4}\times C_{2}$

######  Theorem 1.23 (Main theorem 2).

Let $k$ be a field. Let $T_{i}$ and $T_{j}^{\prime}$ $(1\leq i,j\leq 15)$ be
algebraic $k$-tori of dimension $3$ with the minimal splitting fields $L_{i}$
and $L^{\prime}_{j}$ and the character modules $\widehat{T}_{i}=M_{G}$ and
$\widehat{T}^{\prime}_{j}=M_{G^{\prime}}$ which satisfy that $G$ and
$G^{\prime}$ are ${\rm GL}(3,\mathbbm{Z})$-conjugate to $N_{3,i}$ and
$N_{3,j}$ respectively. For $1\leq i,j\leq 15$, the following conditions are
equivalent:
(1) $T_{i}$ and $T_{j}^{\prime}$ are stably birationally $k$-equivalent;
(2) $L_{i}=L_{j}^{\prime}$, $T_{i}\times_{k}K$ and $T_{j}^{\prime}\times_{k}K$
are weak stably birationally $K$-equivalent for any $k\subset K\subset L_{i}$;
(3) $L_{i}=L_{j}^{\prime}$, $T_{i}\times_{k}K$ and $T_{j}^{\prime}\times_{k}K$
are weak stably birationally $K$-equivalent for any $k\subset K\subset L_{i}$
corresponding to ${\rm WSEC}_{r}$ $(r\geq 1)$;
(4) $L_{i}=L_{j}^{\prime}$, $T_{i}\times_{k}K$ and $T_{j}^{\prime}\times_{k}K$
are weak stably birationally $K$-equivalent for any $k\subset K\subset L_{i}$
corresponding to ${\rm WSEC}_{r}$ $(r\geq 1)$ with $[K:k]=d$ $($the “bold”
cases as in Table $10$$)$ where

$\displaystyle d=\begin{cases}1&(i=1,3,4,8,9,10,11,12,13,14),\\\
1,2&(i=2,5,6,7,15).\end{cases}$

######  Theorem 1.24 (Main theorem 3).

Let $k$ be a field. There exist exactly $129$ weak stably birationally
$k$-equivalent classes of algebraic $k$-tori $T$ of dimension $4$ which
consist of the stably rational class ${\rm WSEC}_{0}$, $121$ classes ${\rm
WSEC}_{r}$ $(1\leq r\leq 121)$ for $[\widehat{T}]^{fl}$ with
$\widehat{T}=M_{G}$ and $G=N_{31,i}$ $(1\leq i\leq 64)$ as in Table $3$ and
for $[\widehat{T}]^{fl}$ with $\widehat{T}=M_{G}$ and $G=N_{4,i}$ $(1\leq
i\leq 152)$ as in Table $4$, and $7$ classes ${\rm WSEC}_{r}$ $(122\leq r\leq
128)$ for $[\widehat{T}]^{fl}$ with $\widehat{T}=M_{G}$ and $G=I_{4,i}$
$(1\leq i\leq 7)$ as in Table $5$. Moreover, the first $13$ weak stably
birationally $k$-equivalent classes ${\rm WSEC}_{r}$ $(1\leq r\leq 13)$ are
common in dimension $3$ $($Theorem 1.22 and Table $2$$)$ and dimension $4$
$($Table $3$ and Table $4$$)$.

Table $3$: Weak stably $k$-equivalent classes for $[\widehat{T}]^{fl}$ with
$\widehat{T}=M_{G}$ and $G=N_{31,i}$ $(1\leq i\leq 64)$

$r$ | $G=N_{31,i}:[\widehat{T}]^{fl}=[M_{G}]^{fl}\in{\rm WSEC}_{r}$ |  $H$: $G=H$ or $G=H\times C_{2}$ |
---|---|---|---
$1$ |  $N_{31,1}$, $N_{31,4}$, $N_{31,11}$ | $C_{2}^{2}$ |
$2$ |  $N_{31,3}$, $N_{31,6}$, $N_{31,8}$, $N_{31,10}$, $N_{31,13}$ | $C_{2}^{3}$ |
$3$ |  $N_{31,2}$, $N_{31,5}$, $N_{31,7}$, $N_{31,9}$, $N_{31,12}$ | $C_{2}^{3}$ |
$4$ |  $N_{31,14}$, $N_{31,18}$, $N_{31,25}$, $N_{31,26}$, $N_{31,29}$ | $C_{4}\times C_{2}$ |
$5$ |  $N_{31,15}$, $N_{31,16}$, $N_{31,19}$, $N_{31,20}$, $N_{31,21}$, $N_{31,22}$, $N_{31,27}$, $N_{31,28}$, $N_{31,31}$, $N_{31,35}$ | $D_{4}$ |
$6$ |  $N_{31,17}$, $N_{31,23}$, $N_{31,24}$, $N_{31,30}$, $N_{31,32}$, $N_{31,33}$, $N_{31,34}$, $N_{31,36}$, $N_{31,37}$ | $D_{4}\times C_{2}$ |
$7$ |  $N_{31,38}$, $N_{31,48}$ | $A_{4}$ |
$8$ |  $N_{31,40}$, $N_{31,47}$, $N_{31,53}$ | $A_{4}\times C_{2}$ |
$9$ |  $N_{31,39}$, $N_{31,46}$, $N_{31,52}$ | $A_{4}\times C_{2}$ |
$10$ |  $N_{31,41}$, $N_{31,43}$, $N_{31,50}$, $N_{31,51}$, $N_{31,60}$, $N_{31,62}$ | $S_{4}$ |
$11$ |  $N_{31,42}$, $N_{31,49}$, $N_{31,61}$ | $S_{4}$ |
$12$ |  $N_{31,45}$, $N_{31,55}$, $N_{31,57}$, $N_{31,59}$, $N_{31,64}$ | $S_{4}\times C_{2}$ |
$13$ |  $N_{31,44}$, $N_{31,54}$, $N_{31,56}$, $N_{31,58}$, $N_{31,63}$ | $S_{4}\times C_{2}$ |

Table $4$: Weak stably $k$-equivalent classes for $[\widehat{T}]^{fl}$ with
$\widehat{T}=M_{G}$ and $G=N_{4,i}$ $(1\leq i\leq 152)$

$r$ | $G=N_{4,i}:[\widehat{T}]^{fl}=[M_{G}]^{fl}\in{\rm WSEC}_{r}$ | $G$
---|---|---
$1$ | $N_{4,1}$ | $C_{2}^{2}$
$2$ | $N_{4,6}$ | $C_{2}^{3}$
$3$ |  $N_{4,2}$, $N_{4,5}$, $N_{4,7}$ | $C_{2}^{3}$
$4$ |  $N_{4,13}$, $N_{4,23}$, $N_{4,24}$ | $C_{4}\times C_{2}$
$5$ |  $N_{4,15}$, $N_{4,16}$, $N_{4,17}$, $N_{4,25}$, $N_{4,26}$ | $D_{4}$
$6$ |  $N_{4,19}$, $N_{4,29}$, $N_{4,30}$ | $D_{4}\times C_{2}$
$7$ |  |
$8$ |  |
$9$ |  $N_{4,69}$, $N_{4,74}$, $N_{4,75}$ | $A_{4}\times C_{2}$
$10$ |  |
$11$ |  $N_{4,71}$, $N_{4,77}$, $N_{4,78}$ | $S_{4}$
$12$ |  |
$13$ |  $N_{4,72}$, $N_{4,83}$, $N_{4,85}$ | $S_{4}\times C_{2}$
$14$ |  $N_{4,3}$, $N_{4,8}$ | $C_{2}^{3}$
$15$ | $N_{4,4}$ | $C_{2}^{3}$
$16$ | $N_{4,9}$ | $C_{2}^{3}$
$17$ | $N_{4,10}$ | $C_{2}^{4}$
$18$ | $N_{4,11}$ | $C_{2}^{4}$
$19$ | $N_{4,12}$ | $C_{2}^{4}$
$20$ |  $N_{4,14}$, $N_{4,39}$ | $C_{4}\times C_{2}$
$21$ | $N_{4,18}$ | $D_{4}$
$22$ |  $N_{4,20}$, $N_{4,31}$, $N_{4,33}$, $N_{4,35}$ | $D_{4}\times C_{2}$
$23$ | $N_{4,21}$ | $D_{4}\times C_{2}$
$24$ |  $N_{4,22}$, $N_{4,46}$ | $D_{4}\times C_{2}$
$25$ | $N_{4,27}$ | $C_{4}\times C_{2}^{2}$
$26$ | $N_{4,28}$ | $C_{4}\times C_{2}^{2}$
$27$ |  $N_{4,32}$, $N_{4,34}$ | $D_{4}\times C_{2}$
$28$ | $N_{4,36}$ | $D_{4}\times C_{2}$
$29$ | $N_{4,37}$ | $D_{4}\times C_{2}^{2}$
$30$ | $N_{4,38}$ | $D_{4}\times C_{2}^{2}$
$31$ | $N_{4,40}$ | $(C_{4}\times C_{2})\rtimes C_{2}$
$32$ |  $N_{4,41}$, $N_{4,44}$ | $(C_{4}\times C_{2})\rtimes C_{2}$
$33$ | $N_{4,42}$ | $(C_{4}\times C_{2})\rtimes C_{2}$
$34$ | $N_{4,43}$ | $(C_{4}\times C_{2})\rtimes C_{2}$
$35$ | $N_{4,45}$ | $D_{4}\times C_{2}$
$36$ | $N_{4,47}$ | $C_{2}^{4}\rtimes C_{2}$
$37$ | $N_{4,48}$ | $C_{2}^{4}\rtimes C_{2}$
$38$ | $N_{4,49}$ | $C_{2}^{4}\rtimes C_{2}$
$39$ | $N_{4,50}$ | $C_{4}\rtimes C_{4}$
$40$ | $N_{4,51}$ | $C_{4}^{2}$
$41$ | $N_{4,52}$ | $D_{4}\times C_{4}$
$42$ | $N_{4,53}$ | $(C_{4}\times C_{2}^{2})\rtimes C_{2}$
$43$ | $N_{4,54}$ | $(C_{4}\times C_{2}^{2})\rtimes C_{2}$
$44$ | $N_{4,55}$ | $C_{4}^{2}\rtimes C_{2}$
$45$ | $N_{4,56}$ | $D_{4}^{2}$
$46$ | $N_{4,57}$ | $C_{3}^{2}$
$47$ | $N_{4,58}$ | $C_{6}\times C_{3}$
$48$ |  $N_{4,59}$, $N_{4,91}$ | $S_{3}\times C_{3}$
$49$ |  $N_{4,60}$, $N_{4,92}$ | $S_{3}\times C_{3}$
$50$ | $N_{4,61}$ | $C_{3}^{2}\rtimes C_{2}$
$51$ | $N_{4,62}$ | $C_{3}^{2}\rtimes C_{2}$
$52$ |  $N_{4,63}$, $N_{4,93}$ | $S_{3}\times C_{6}$
$53$ | $N_{4,64}$ | $(C_{3}^{2}\rtimes C_{2})\times C_{2}$
$54$ |  $N_{4,65}$, $N_{4,94}$ | $S_{3}^{2}$
$55$ |  $N_{4,66}$, $N_{4,96}$ | $S_{3}^{2}$
$56$ |  $N_{4,67}$, $N_{4,95}$ | $S_{3}^{2}$
$57$ |  $N_{4,68}$, $N_{4,99}$ | $S_{3}^{2}\times C_{2}$
$58$ | $N_{4,70}$ | $A_{4}\times C_{2}$
$59$ | $N_{4,73}$ | $S_{4}\times C_{2}$
$60$ | $N_{4,76}$ | $A_{4}\times C_{2}$
$61$ | $N_{4,79}$ | $A_{4}\times C_{2}^{2}$
$62$ | $N_{4,80}$ | $A_{4}\times C_{2}^{2}$
$63$ |  $N_{4,81}$, $N_{4,84}$, $N_{4,87}$ | $S_{4}\times C_{2}$
$64$ | $N_{4,82}$ | $S_{4}\times C_{2}$
$65$ |  $N_{4,86}$, $N_{4,88}$ | $S_{4}\times C_{2}$
$66$ | $N_{4,89}$ | $S_{4}\times C_{2}^{2}$
$67$ | $N_{4,90}$ | $S_{4}\times C_{2}^{2}$
$68$ | $N_{4,97}$ | $C_{3}^{2}\rtimes C_{4}$
$69$ | $N_{4,98}$ | $C_{3}^{2}\rtimes C_{4}$
$70$ | $N_{4,100}$ | $(C_{3}^{2}\rtimes C_{4})\times C_{2}$
$71$ | $N_{4,101}$ | $S_{3}^{2}\rtimes C_{2}$
$72$ |  $N_{4,102}$, $N_{4,103}$ | $S_{3}^{2}\rtimes C_{2}$
$73$ | $N_{4,104}$ | $S_{3}^{2}\rtimes C_{2}$
$74$ | $N_{4,105}$ | $(S_{3}^{2}\rtimes C_{2})\times C_{2}$
$75$ | $N_{4,106}$ | $Q_{8}$
$76$ | $N_{4,107}$ | $C_{8}\rtimes C_{2}$
$77$ | $N_{4,108}$ | $QD_{8}$
$78$ | $N_{4,109}$ | $(C_{4}\times C_{2})\rtimes C_{2}$
$79$ | $N_{4,110}$ | ${\rm SL}(2,\mathbbm{F}_{3})$
$80$ | $N_{4,111}$ | ${\rm SL}(2,\mathbbm{F}_{3})$
$81$ | $N_{4,112}$ | $C_{8}\rtimes C_{2}^{2}$
$82$ | $N_{4,113}$ | $(C_{8}\rtimes C_{2})\rtimes C_{2}$
$83$ | $N_{4,114}$ | $C_{4}^{2}\rtimes C_{2}$
$84$ | $N_{4,115}$ | $C_{2}^{3}\rtimes C_{4}$
$85$ | $N_{4,116}$ | $C_{2}^{3}\rtimes C_{4}$
$86$ | $N_{4,117}$ | $C_{2}^{3}\rtimes C_{2}^{2}$
$87$ | $N_{4,118}$ | ${\rm GL}(2,\mathbbm{F}_{3})$
$88$ | $N_{4,119}$ | ${\rm GL}(2,\mathbbm{F}_{3})$
$89$ | $N_{4,120}$ | $((C_{8}\rtimes C_{2})\rtimes C_{2})\rtimes C_{2}$
$90$ | $N_{4,121}$ | $((C_{8}\rtimes C_{2})\rtimes C_{2})\rtimes C_{2}$
$91$ | $N_{4,122}$ | $((C_{8}\rtimes C_{2})\rtimes C_{2})\rtimes C_{2}$
$92$ | $N_{4,123}$ | $(C_{2}^{4}\rtimes C_{2})\rtimes C_{2}$
$93$ | $N_{4,124}$ | $(C_{2}^{4}\rtimes C_{2})\rtimes C_{2}$
$94$ | $N_{4,125}$ | $(C_{2}^{3}\rtimes C_{4})\rtimes C_{2}$
$95$ | $N_{4,126}$ | $(C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3}$
$96$ | $N_{4,127}$ | $(C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3}$
$97$ | $N_{4,128}$ | $D_{4}^{2}\rtimes C_{2}$
$98$ | $N_{4,129}$ | $((C_{2}^{4}\rtimes C_{2})\rtimes C_{2})\rtimes C_{3}$
$99$ | $N_{4,130}$ | $((C_{2}^{4}\rtimes C_{2})\rtimes C_{2})\rtimes C_{3}$
$100$ | $N_{4,131}$ | $((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2}$
$101$ | $N_{4,132}$ | $((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2}$
$102$ | $N_{4,133}$ | $((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2}$
$103$ | $N_{4,134}$ | $((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2}$
$104$ | $N_{4,135}$ | $(((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2})\rtimes C_{2}$
$105$ | $N_{4,136}$ | $(((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2})\rtimes C_{2}$
$106$ | $N_{4,137}$ | $Q_{8}\times C_{3}$
$107$ | $N_{4,138}$ | ${\rm SL}(2,\mathbbm{F}_{3})$
$108$ | $N_{4,139}$ | $(Q_{8}\times C_{3})\rtimes C_{2}$
$109$ | $N_{4,140}$ | $((C_{4}\times C_{2})\rtimes C_{2})\rtimes C_{3}$
$110$ | $N_{4,141}$ | ${\rm GL}(2,\mathbbm{F}_{3})$
$111$ | $N_{4,142}$ | ${\rm SL}(2,\mathbbm{F}_{3})\times C_{3}$
$112$ | $N_{4,143}$ | $(C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3}$
$113$ | $N_{4,144}$ | ${\rm GL}(2,\mathbbm{F}_{3})\rtimes C_{2}$
$114$ | $N_{4,145}$ | ${\rm SL}(2,\mathbbm{F}_{3})\rtimes C_{4}$
$115$ | $N_{4,146}$ | $({\rm SL}(2,\mathbbm{F}_{3})\times C_{3})\rtimes C_{2}$
$116$ | $N_{4,147}$ | $({\rm GL}(2,\mathbbm{F}_{3})\rtimes C_{2})\rtimes C_{2}$
$117$ | $N_{4,148}$ | $(C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3}^{2}$
$118$ | $N_{4,149}$ | $(((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2})\rtimes C_{3}$
$119$ | $N_{4,150}$ | $(((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2})\rtimes C_{3}$
$120$ | $N_{4,151}$ | $((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3}^{2})\rtimes C_{2}$
$121$ | $N_{4,152}$ | $(((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3}^{2})\rtimes C_{2})\rtimes C_{2}$

Table $5$: Weak stably $k$-equivalent classes for $[\widehat{T}]^{fl}$ with
$\widehat{T}=M_{G}$ and $G=I_{4,i}$ $(1\leq i\leq 7)$

$r$ | $G=I_{4,i}:[\widehat{T}]^{fl}=[M_{G}]^{fl}\in{\rm WSEC}_{r}$ | $G$
---|---|---
$122$ | $I_{4,1}$ | $F_{20}$
$123$ | $I_{4,2}$ | $F_{20}$
$124$ | $I_{4,3}$ | $F_{20}\times C_{2}$
$125$ | $I_{4,4}$ | $S_{5}$
$126$ | $I_{4,5}$ | $S_{5}$
$127$ | $I_{4,6}$ | $S_{5}\times C_{2}$
$128$ | $I_{4,7}$ | $C_{3}\rtimes C_{8}$

######  Remark 1.25.

For $G=N_{3,i}$, $N_{4,i}$ or $I_{4,i}$, and $G^{\prime}=N_{3,j}$, $N_{4,j}$
or $I_{4,j}$, it follows from Theorem 1.22 and Theorem 1.24 that if
$[M_{G}]^{fl}\sim[M_{G^{\prime}}]^{fl}$, then $G\simeq G^{\prime}$. However,
in general, it occurs that $[M_{G}]^{fl}\sim[M_{G^{\prime}}]^{fl}$ and
$G\not\simeq G^{\prime}$ (see Table $3$). We found that there exists
$G^{\prime\prime}\leq{\rm GL}(5,\mathbbm{Z})$ such that
$[M_{G^{\prime\prime}}]^{fl}\sim[M_{G}]^{fl}$ with indecomposable
$G^{\prime\prime}$-lattice $M_{G^{\prime\prime}}$, $G^{\prime\prime}\simeq
D_{4}$ with CARAT ID (5,100,12) (see Hoshi and Yamasaki [HY17, Chapter 3]) and
$G=N_{3,1}\simeq C_{2}^{2}$.

Let ${\rm Aut}(G)$ be the group of automorphisms on $G$ and ${\rm Inn}(G)$ be
the group of inner automorphisms on $G$.

######  Theorem 1.26 (Main theorem 4).

Let $k$ be a field. Let $T_{i}$ and $T_{j}^{\prime}$ $(1\leq i,j\leq 152)$ be
algebraic $k$-tori of dimension $4$ with the minimal splitting fields $L_{i}$
and $L^{\prime}_{j}$ and the character modules $\widehat{T}_{i}=M_{G}$ and
$\widehat{T}^{\prime}_{j}=M_{G^{\prime}}$ which satisfy that $G$ and
$G^{\prime}$ are ${\rm GL}(4,\mathbbm{Z})$-conjugate to $N_{4,i}$ and
$N_{4,j}$ respectively. For $1\leq i,j\leq 152$ except for the cases
$i=j=137,139,145,147$, the following conditions are equivalent:
(1) $T_{i}$ and $T_{j}^{\prime}$ are stably birationally $k$-equivalent;
(2) $L_{i}=L_{j}^{\prime}$, $T_{i}\times_{k}K$ and $T_{j}^{\prime}\times_{k}K$
are weak stably birationally $K$-equivalent for any $k\subset K\subset L_{i}$;
(3) $L_{i}=L_{j}^{\prime}$, $T_{i}\times_{k}K$ and $T_{j}^{\prime}\times_{k}K$
are weak stably birationally $K$-equivalent for any $k\subset K\subset L_{i}$
corresponding to ${\rm WSEC}_{r}$ $(r\geq 1)$;
(4) $L_{i}=L_{j}^{\prime}$, $T_{i}\times_{k}K$ and $T_{j}^{\prime}\times_{k}K$
are weak stably birationally $K$-equivalent for any $k\subset K\subset L_{i}$
corresponding to ${\rm WSEC}_{r}$ $(r\geq 1)$ with $[K:k]=d$ $($the “bold”
cases as in Table $12$$)$ where

$\displaystyle
d=\begin{cases}1&(i=1,2,4,5,7,10,13,14,23,24,27,39,41,42,44,49,57,58,59,60,61,62,65,67,69,70,71,73,\\\
&\hskip
19.91692pt74,75,76,77,78,79,81,82,84,87,91,92,94,95,97,98,101,104,106,107,108,109,110,111,\\\
&\hskip
19.91692pt114,115,116,118,119,121,124,126,127,130,132,133,134,138,137,139,140,141,145),\\\
1,2&(i=3,6,8,9,11,12,15,16,17,18,19,20,22,25,26,29,30,31,33,35,37,43,46,50,51,54,55,\\\
&\hskip
19.91692pt63,64,66,68,72,80,83,85,89,90,93,96,99,100,102,103,105,125),\\\
1,3&(i=142,148),\\\ 1,4&(i=21,36,38,45,47,48,52,53,56,113,117,120),\\\
1,2,4&(i=28,32,34,40),\\\ 1,6&(i=86,88),\\\
1,8&(i=112,122,123,128,129,131,135,136),\\\ 1,3,8&(i=146),\\\
1,12&(i=143,147),\\\ 1,24&(i=144,149),\\\ 1,32&(i=150),\\\ 1,2,32&(i=152),\\\
1,3,32&(i=151).\end{cases}$

For the exceptional cases $i=j=137,139,145,147$ $($$G\simeq Q_{8}\times
C_{3}$, $(Q_{8}\times C_{3})\rtimes C_{2}$, ${\rm
SL}(2,\mathbbm{F}_{3})\rtimes C_{4}$, $({\rm GL}(2,\mathbbm{F}_{3})\rtimes
C_{2})\rtimes C_{2}\simeq({\rm SL}(2,\mathbbm{F}_{3})\rtimes C_{4})\rtimes
C_{2}$$)$, we have the implications
${\rm(1)}\Rightarrow{\rm(2)}\Leftrightarrow{\rm(3)}\Leftrightarrow{\rm(4)}$,
there exists $\tau\in{\rm Aut}(G)$ such that $G^{\prime}=G^{\tau}$ and
$X=Y\lhd Z$ with $Z/Y\simeq C_{2},C_{2}^{2},C_{2},C_{2}$ respectively where

${\rm Inn}(G)\leq X\leq Y\leq Z\leq{\rm Aut}(G)$,

$\displaystyle X$ $\displaystyle={\rm Aut}_{{\rm
GL}(4,\mathbbm{Z})}(G)=\\{\sigma\in{\rm Aut}(G)\mid G\ {\rm and}\ G^{\sigma}\
{\rm are}\ {\rm conjugate}\ {\rm in}\ {\rm GL}(4,\mathbbm{Z})\\},$
$\displaystyle Y$ $\displaystyle=\\{\sigma\in{\rm
Aut}(G)\mid[M_{G}]^{fl}=[M_{G^{\sigma}}]^{fl}\ {\rm as}\
\widetilde{H}\textrm{-{\rm lattices}}\ {\rm where}\
\widetilde{H}=\\{(g,g^{\sigma})\mid g\in G\\}\simeq G\\},$ $\displaystyle Z$
$\displaystyle=\\{\sigma\in{\rm
Aut}(G)\mid[M_{H}]^{fl}\sim[M_{H^{\sigma}}]^{fl}\ {\rm for}\ {\rm any}\ H\leq
G\\}.$

Moreover, we have ${\rm(1)}$ $\Leftrightarrow M_{G}\simeq M_{G^{\tau}}$ as
$\widetilde{H}$-lattices $\Leftrightarrow
M_{G}\otimes_{\mathbbm{Z}}\mathbbm{F}_{p}\simeq
M_{G^{\tau}}\otimes_{\mathbbm{Z}}\mathbbm{F}_{p}$ as
$\mathbbm{F}_{p}[\widetilde{H}]$-lattices for

$\displaystyle p=\begin{cases}2&(i=j=137),\\\ 2\ \,{\rm and}\
\,3&(i=j=139),\\\ 3&(i=j=145,147).\end{cases}$

######  Remark 1.27.

(1) Under the assumption $L_{i}=L^{\prime}_{j}$, i.e. $\widetilde{H}\simeq
G\simeq G^{\prime}$, ${\rm(1)}\Leftrightarrow{\rm(2)}$ of Theorem 1.26 is
equivalent to $Y=Z$. In particular, if ${\rm Inn}(G)={\rm Aut}(G)$, then
${\rm(1)}\Leftrightarrow{\rm(2)}$ of Theorem 1.26 holds.
(2) We give the condition of Theorem 1.26 (4) for $d=[K:k]\in l$ as the list
$l=[l_{1},\ldots,l_{s}]$ $(l_{1}\leq\cdots\leq l_{s})$ of integers with the
minimal $l_{s},\ldots,l_{1}$ (see Section 10).
(3) In Theorem 1.26 (4), ${\rm Aut}(G)$ may permute subgroups
$H,H^{\prime}\leq G$ with $K=L_{i}^{H}$, $K^{\prime}=L_{i}^{H^{\prime}}$
corresponding to ${\rm WSEC}_{0}$, ${\rm WSEC}_{r}$ $(r\geq 1)$ respectively.

######  Example 1.28.

(1) For the exceptional case $i=j=137$ $($$G\simeq Q_{8}\times C_{3}$$)$ in
Theorem 1.26, we have
$G=N_{4,137}=\langle\lambda_{1},\lambda_{2}\mid\lambda_{1}^{12}=I,\lambda_{2}^{2}=\lambda_{1}^{6},[\lambda_{1},\lambda_{2}]=\lambda_{1}^{6}\rangle$
with GAP ID $(4,33,1,1)$ (see Table $8$ and [BBNWZ78, Table 1C, page 257]),
$G^{\tau}=\langle\lambda_{1}^{\tau},\lambda_{2}^{\tau}\rangle$ and
$Z/Y=\\{{\rm id}_{G},\tau\\}$ where

$\displaystyle\lambda_{1}=\lambda_{1}^{\tau}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}0&1&0&-1\\\
0&0&-1&1\\\ -1&0&0&1\\\
0&1&0&0\end{array}$}}\right),\lambda_{2}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}0&0&-1&0\\\
-1&-1&-1&2\\\ 1&0&0&0\\\
0&-1&-1&1\end{array}$}}\right),\lambda_{2}^{\tau}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}0&1&0&0\\\
-1&0&0&0\\\ -1&-1&-1&2\\\ -1&0&-1&1\end{array}$}}\right).$

(2) For the exceptional case $i=j=139$ $($$G\simeq(Q_{8}\times C_{3})\rtimes
C_{2}$$)$ in Theorem 1.26, we have
$G=N_{4,137}=\langle\lambda_{1},\lambda_{2},\lambda_{3}\mid\lambda_{1}^{12}=I,\lambda_{2}^{2}=\lambda_{1}^{6},\lambda_{3}^{2}=I,[\lambda_{1},\lambda_{2}]=\lambda_{1}^{6},[\lambda_{1},\lambda_{3}]=\lambda_{1}^{10},[\lambda_{2},\lambda_{3}]=\lambda_{1}^{9}\rangle$
with GAP ID $(4,33,4,1)$ (see Table $8$ and [BBNWZ78, Table 1C, page 257]) and
$Z/Y=\\{{\rm id}_{G},\tau,\tau^{\prime},\tau\tau^{\prime}\\}$ where
$\lambda_{1}=\lambda_{1}^{\tau}$, $\lambda_{2}$ and $\lambda_{2}^{\tau}$ are
given as in (1),

$\displaystyle\lambda_{3}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}1&0&0&0\\\
0&0&-1&0\\\ 0&-1&0&0\\\
1&0&0&-1\end{array}$}}\right),\lambda_{3}^{\tau}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}-1&-1&-1&2\\\
0&1&0&0\\\ 0&0&-1&0\\\
0&0&-1&1\end{array}$}}\right),\lambda_{1}^{\tau^{\prime}}=\lambda_{1}^{\tau\tau^{\prime}}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}1&0&1&-1\\\
0&0&0&-1\\\ 1&1&0&-1\\\ 1&0&0&-1\end{array}$}}\right),$

$\lambda_{2}^{\tau^{\prime}}=\lambda_{2}$,
$\lambda_{2}^{\tau\tau^{\prime}}=\lambda_{2}^{\tau}$,
$\lambda_{3}^{\tau^{\prime}}=-\lambda_{3}$,
$\lambda_{3}^{\tau\tau^{\prime}}=-\lambda_{3}^{\tau}$.
(3) For the exceptional case $i=j=145$ $($$G\simeq{\rm
SL}(2,\mathbbm{F}_{3})\rtimes C_{4}$$)$ in Theorem 1.26, we have
$G=N_{4,137}=\langle\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\mid\alpha_{1}^{4}=I,\alpha_{2}^{2}=\alpha_{1}^{2},\alpha_{3}^{3}=I,\alpha_{4}^{4}=I,[\alpha_{1},\alpha_{2}]=\alpha_{1}^{2},[\alpha_{1},\alpha_{3}]=\alpha_{2}\alpha_{1}^{2},[\alpha_{1},\alpha_{4}]=I,[\alpha_{2},\alpha_{3}]=\alpha_{2}\alpha_{1},[\alpha_{2},\alpha_{4}]=\alpha_{1},[\alpha_{3},\alpha_{4}]=\alpha_{3}\alpha_{2}\alpha_{1}^{2}\rangle$
with GAP ID $(4,33,10,1)$ (see Table $8$ and [BBNWZ78, Table 1C, page 258])
and $Z/Y=\\{{\rm id}_{G},\tau\\}$ where

$\displaystyle\alpha_{1}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}-1&-1&-1&2\\\
0&0&1&0\\\ 0&-1&0&0\\\
-1&-1&0&1\end{array}$}}\right),\alpha_{2}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}0&0&-1&0\\\
-1&-1&-1&2\\\ 1&0&0&0\\\
0&-1&-1&1\end{array}$}}\right),\alpha_{3}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}0&0&0&-1\\\
0&-1&-1&1\\\ 1&1&0&-1\\\
1&0&0&-1\end{array}$}}\right),\alpha_{4}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}-1&-1&-1&2\\\
0&1&0&0\\\ 0&0&1&0\\\ -1&0&0&1\end{array}$}}\right),$

$\alpha_{1}^{\tau}=\alpha_{1}$, $\alpha_{2}^{\tau}=\alpha_{2}$,
$\alpha_{3}^{\tau}=\alpha_{3}$, $\alpha_{4}^{\tau}=-\alpha_{4}$.
(4) For the exceptional case $i=j=147$ $($$G\simeq({\rm
GL}(2,\mathbbm{F}_{3})\rtimes C_{2})\rtimes C_{2}\simeq({\rm
SL}(2,\mathbbm{F}_{3})\rtimes C_{4})\rtimes C_{2}$$)$ in Theorem 1.26, we have
$G=N_{4,137}=\langle\beta_{1},\beta_{2},\beta_{3},\beta_{4},\beta_{5}\mid\beta_{1}^{2}=I,\beta_{2}^{4}=I,\beta_{3}^{2}=I,\beta_{4}^{6}=\beta_{2}^{2},\beta_{5}^{2}=I,[\beta_{1},\beta_{2}]=I,[\beta_{1},\beta_{3}]=\beta_{2}^{2},[\beta_{1},\beta_{4}]=\beta_{3}\beta_{2}^{3},[\beta_{1},\beta_{5}]=I,[\beta_{2},\beta_{3}]=I,[\beta_{2},\beta_{4}]=\beta_{2}^{2},[\beta_{2},\beta_{5}]=\beta_{2}^{2},[\beta_{3},\beta_{4}]=\beta_{3}\beta_{2}^{2}\beta_{1},[\beta_{3},\beta_{5}]=\beta_{2}\beta_{1},[\beta_{4},\beta_{5}]=\beta_{4}^{4}\beta_{3}\beta_{2}\beta_{1}\rangle$
with GAP ID $(4,33,12,1)$ (see Table $8$ and [BBNWZ78, Table 1C, page 259])
and $Z/Y=\\{{\rm id}_{G},\tau\\}$ where

$\displaystyle\beta_{1}$
$\displaystyle=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}1&0&0&0\\\
0&-1&0&0\\\ 0&0&-1&0\\\
0&-1&-1&1\end{array}$}}\right),\beta_{2}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}-1&-1&-1&2\\\
0&0&1&0\\\ 0&-1&0&0\\\
-1&-1&0&1\end{array}$}}\right),\beta_{3}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}0&0&-1&0\\\
-1&-1&-1&2\\\ -1&0&0&0\\\
-1&0&-1&1\end{array}$}}\right),\beta_{4}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}-1&0&-1&1\\\
0&0&0&-1\\\ -1&-1&0&1\\\ -1&-1&-1&1\end{array}$}}\right),$
$\displaystyle\beta_{5}$
$\displaystyle=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}1&0&0&0\\\
0&0&1&0\\\ 0&1&0&0\\\
1&1&1&-1\end{array}$}}\right),\beta_{1}^{\tau}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}0&0&1&0\\\
1&1&1&-2\\\ 1&0&0&0\\\
1&0&1&-1\end{array}$}}\right),\beta_{3}^{\tau}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}0&-1&0&0\\\
-1&0&0&0\\\ 1&1&1&-2\\\
0&0&0&-1\end{array}$}}\right),\beta_{5}^{\tau}\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}0&-1&-1&1\\\
0&0&0&-1\\\ -1&-1&0&1\\\ 0&-1&0&0\end{array}$}}\right),$

$\beta_{2}^{\tau}=-\beta_{2}$, $\beta_{4}^{\tau}=\beta_{4}$.

######  Theorem 1.29 (Main theorem 5).

Let $k$ be a field. Let $T_{i}$ and $T_{j}^{\prime}$ $(1\leq i,j\leq 7)$ be
algebraic $k$-tori of dimension $4$ with the minimal splitting fields $L_{i}$
and $L^{\prime}_{j}$ and the character modules $\widehat{T}_{i}=M_{G}$ and
$\widehat{T}^{\prime}_{j}=M_{G^{\prime}}$ which satisfy that $G$ and
$G^{\prime}$ are ${\rm GL}(4,\mathbbm{Z})$-conjugate to $I_{4,i}$ and
$I_{4,j}$ respectively. For $1\leq i,j\leq 7$ except for the case $i=j=7$, the
following conditions are equivalent:
(1) $T_{i}$ and $T_{j}^{\prime}$ are stably birationally $k$-equivalent;
(2) $L_{i}=L_{j}^{\prime}$, $T_{i}\times_{k}K$ and $T_{j}^{\prime}\times_{k}K$
are weak stably birationally $K$-equivalent for any $k\subset K\subset L_{i}$;
(3) $L_{i}=L_{j}^{\prime}$, $T_{i}\times_{k}K$ and $T_{j}^{\prime}\times_{k}K$
are weak stably birationally $K$-equivalent for any $k\subset K\subset L_{i}$
corresponding to ${\rm WSEC}_{r}$ $(r\geq 1)$;
(4) $L_{i}=L_{j}^{\prime}$, $T_{i}\times_{k}K$ and $T_{j}^{\prime}\times_{k}K$
are weak stably birationally $K$-equivalent for any $k\subset K\subset L_{i}$
corresponding to ${\rm WSEC}_{r}$ $(r\geq 1)$ with $[K:k]=d$ $($the “bold”
cases as in Table $13$$)$ where

$\displaystyle d=\begin{cases}1&(i=1,2,4,5,7),\\\ 1,2&(i=3,6).\end{cases}$

For the exceptional case $i=j=7$ $($$G\simeq C_{3}\rtimes C_{8}$$)$, we have
the implications
${\rm(1)}\Rightarrow{\rm(2)}\Leftrightarrow{\rm(3)}\Leftrightarrow{\rm(4)}$,
there exists $\tau\in{\rm Aut}(G)$ such that $G^{\prime}=G^{\tau}$ and
$X=Y\lhd Z$ with $Z/Y\simeq C_{2}$ where

${\rm Inn}(G)\simeq S_{3}\leq X\leq Y\leq Z\leq{\rm Aut}(G)\simeq S_{3}\times
C_{2}^{2}$,

$\displaystyle X$ $\displaystyle={\rm Aut}_{{\rm
GL}(4,\mathbbm{Z})}(G)=\\{\sigma\in{\rm Aut}(G)\mid G\ {\rm and}\ G^{\sigma}\
{\rm are}\ {\rm conjugate}\ {\rm in}\ {\rm GL}(4,\mathbbm{Z})\\}\simeq D_{6},$
$\displaystyle Y$ $\displaystyle=\\{\sigma\in{\rm
Aut}(G)\mid[M_{G}]^{fl}=[M_{G^{\sigma}}]^{fl}\ {\rm as}\
\widetilde{H}\textrm{-{\rm lattices}}\ {\rm where}\
\widetilde{H}=\\{(g,g^{\sigma})\mid g\in G\\}\simeq G\\}\simeq D_{6},$
$\displaystyle Z$ $\displaystyle=\\{\sigma\in{\rm
Aut}(G)\mid[M_{H}]^{fl}\sim[M_{H^{\sigma}}]^{fl}\ {\rm for}\ {\rm any}\ H\leq
G\\}\simeq S_{3}\times C_{2}^{2}.$

Moreover, we have ${\rm(1)}$ $\Leftrightarrow M_{G}\simeq M_{G^{\tau}}$ as
$\widetilde{H}$-lattices $\Leftrightarrow
M_{G}\otimes_{\mathbbm{Z}}\mathbbm{F}_{3}\simeq
M_{G^{\tau}}\otimes_{\mathbbm{Z}}\mathbbm{F}_{3}$ as
$\mathbbm{F}_{3}[\widetilde{H}]$-lattices.

######  Example 1.30.

For the exceptional case $i=j=7$ $($$G\simeq C_{3}\rtimes C_{8}$$)$ in Theorem
1.29, we have
$G=I_{4,7}=\langle\lambda_{1},\lambda_{4}\mid\lambda_{1}^{12}=I,\lambda_{4}^{2}=\lambda_{1}^{9},[\lambda_{1},\lambda_{4}]=\lambda_{1}^{4}\rangle$
with GAP ID $(4,33,2,1)$ (see Table $9$ and [BBNWZ78, Table 1C, page 257]) and
$Z/Y=\\{{\rm id}_{G},\tau\\}$ where

$\displaystyle\lambda_{1}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}0&1&0&-1\\\
0&0&-1&1\\\ -1&0&0&1\\\
0&1&0&0\end{array}$}}\right),\lambda_{4}=\left(\scalebox{0.9}{\mbox{$\displaystyle\begin{array}[]{cccc}0&0&-1&0\\\
-1&0&0&0\\\ 1&1&1&-2\\\ 0&1&0&-1\end{array}$}}\right),$

$\lambda_{1}^{\tau}=\lambda_{1}$, $\lambda_{4}^{\tau}=-\lambda_{4}$.

For algebraic $k$-tori $T_{i}$ $(1\leq i\leq 7)$ as in Theorem 1.29, we find
the following relationships among them (the statements (1) and (3) of Theorem
1.31 were proved in Hoshi and Yamasaki [HY17, Theorem 1.27] and we will give
an alternative proof of them, see also Remark 1.19 (2)):

######  Theorem 1.31 (Main theorem 6).

Let $k$ be a field. Let $T_{i}$ $(1\leq i,j\leq 7)$ be an algebraic $k$-torus
of dimension $4$ with the character module $\widehat{T}_{i}=M_{G}$ which
satisfies that $G$ is ${\rm GL}(4,\mathbbm{Z})$-conjugate to $I_{4,i}$. Let
$T_{i}^{\sigma}$ be the algebraic $k$-torus with
$\widehat{T}_{i}^{\sigma}=M_{G^{\sigma}}$ $(\sigma\in{\rm Aut}(G))$. Then
$T_{i}$ and $T_{i}^{\sigma}$ are not stably $k$-rational but we have:
(1) $T_{1}\times_{k}T_{2}$ is stably $k$-rational;
(2) $T_{3}\times_{k}T_{3}^{\sigma}$ is stably $k$-rational for $\sigma\in{\rm
Aut}(G)$ with $1\neq\overline{\sigma}\in{\rm Aut}(G)/{\rm Inn}(G)\simeq
C_{2}$;
(3) $T_{4}\times_{k}T_{5}$ is stably $k$-rational;
(4) $T_{6}\times_{k}T_{6}^{\sigma}$ is stably $k$-rational for $\sigma\in{\rm
Aut}(G)$ with $1\neq\overline{\sigma}\in{\rm Aut}(G)/{\rm Inn}(G)\simeq
C_{2}$;
(5) $T_{7}\times_{k}T_{7}^{\sigma}$ is stably $k$-rational for $\sigma\in{\rm
Aut}(G)$ with $1\neq\overline{\sigma}\in{\rm Aut}(G)/X\simeq C_{2}$ where

$\displaystyle X={\rm Aut}_{{\rm GL}(4,\mathbbm{Z})}(G)=\\{\sigma\in{\rm
Aut}(G)\mid G\ {\rm and}\ G^{\sigma}\ {\rm are}\ {\rm conjugate}\ {\rm in}\
{\rm GL}(4,\mathbbm{Z})\\}\simeq D_{6}.$

The following theorem can answer Problem 1.3 for algebraic $k$-tori $T$ and
$T^{\prime}$ of dimensions $m\geq 3$ and $n\geq 3$ respectively with
$[\widehat{T}]^{fl},[\widehat{T}^{\prime}]^{fl}\in{\rm WSEC}_{r}$ $(1\leq
r\leq 128)$ via some algebraic $k$-torus $T^{\prime\prime}$ with dim
$T^{\prime\prime}=3$ or $4$ as in Theorem 1.23, Theorem 1.26 or Theorem 1.29.

######  Theorem 1.32 (Main theorem 7).

Let $k$ be a field. Let $T$ be an algebraic $k$-torus of dimension $m\geq 3$
with the minimal splitting field $L$, $\widehat{T}=M_{G}$, $G\leq{\rm
GL}(m,\mathbbm{Z})$ and $[\widehat{T}]^{fl}\in{\rm WSEC}_{r}$ $(1\leq r\leq
128)$. Then there exists an algebraic $k$-torus $T^{\prime\prime}$ of
dimension $3$ or $4$ with the minimal splitting field $L^{\prime\prime}$,
$\widehat{T}^{\prime\prime}=M_{G^{\prime\prime}}$, and
$G^{\prime\prime}=N_{3,i}$ $(1\leq i\leq 15)$, $G^{\prime\prime}=N_{4,i}$
$(1\leq i\leq 152)$ or $G^{\prime\prime}=I_{4,i}$ $(1\leq i\leq 7)$ such that
$T^{\prime\prime}$ and $T$ are stably birationally $k$-equivalent and
$L^{\prime\prime}\subset L$, i.e. $[M_{G^{\prime\prime}}]^{fl}=[M_{G}]^{fl}$
as $G$-lattices and $G$ acts on $[M_{G^{\prime\prime}}]^{fl}$ though
$G^{\prime\prime}\simeq G/N$ for the corresponding normal subgroup $N\lhd G$.

######  Remark 1.33.

(1) It follows from Theorem Theorem 1.23, Theorem 1.26 and Theorem 1.29 that
the field $L^{\prime\prime}$ (the normal subgroup $N\lhd G$) can be uniquely
determined by $T$. For example, consider $G=N_{31,i}$ (see Table $3$).
(2) Theorem 1.32 does not hold for $G^{\prime\prime}=N_{31,i}$. For example,
take $G=N_{31,11}\simeq G^{\prime\prime}=N_{31,11}\simeq C_{2}^{3}$. Then it
may occur that $[L\cap L^{\prime\prime}:k]=4$ and
$[M_{G}]^{fl}=[M_{G^{\prime\prime}}]^{fl}$ as $\widetilde{H}$-lattices where
$\widetilde{H}={\rm Gal}(LL^{\prime\prime}/k)$ and $G/N\simeq
G^{\prime\prime}/N^{\prime\prime}\simeq N_{3,1}\simeq C^{2}$ for some normal
subgroups $N\lhd G$, $N^{\prime\prime}\lhd G^{\prime\prime}$. Similarly,
Theorem 1.32 does not hold for dimension $5$. Indeed, we can take
$G,G^{\prime\prime}\leq{\rm GL}(5,\mathbbm{Z})$ with CARAT ID (5,100,12),
indecomposable $G$-lattice $M_{G}$, indecomposable $G^{\prime\prime}$-lattice
$M_{G^{\prime\prime}}$, $G\simeq G^{\prime\prime}\simeq D_{4}$, $[L\cap
L^{\prime\prime}:k]=4$ and $[M_{G}]^{fl}=[M_{G^{\prime\prime}}]^{fl}$ as
$\widetilde{H}$-lattices where $\widetilde{H}={\rm Gal}(LL^{\prime\prime}/k)$
and $G/N\simeq G^{\prime\prime}/N^{\prime\prime}\simeq N_{3,1}\simeq C^{2}$
(see Remark 1.25).
(3) If $G\leq{\rm GL}(m,\mathbbm{Z})$ has no normal subgroup $N$ which
satisfies that $G/N$ is in Table $6$ ($G=N_{3,i}$), Table $8$ ($G=N_{4,i}$) or
Table $9$ ($G=I_{4,i}$), then $[\widehat{T}]^{fl}\not\in{\rm WSEC}_{r}$
$(1\leq r\leq 128)$. For example, consider the norm one torus
$T=R^{(1)}_{K/k}$ of dimension $n-1$ with $[K:k]=n$ and $G={\rm
Gal}(L/k)\simeq A_{n}\leq{\rm GL}(n-1,\mathbbm{Z})$ $(n\geq 6)$ where $L$ is
the Galois closure of $K/k$ (see Endo [End11, Theorem 4.4], Hoshi and Yamasaki
[HY17, Corollary 1.11]).

We organize this paper as follows. In Section 2, the groups $N_{3,i}$,
$N_{31,i}$, $N_{4,i}$ and $I_{4,i}$ in Definition 1.18 are given as in Tables
$6,7,8,9$. In Section 3, all of the subgroups $N_{3,j}\leq N_{3,i}$ $(1\leq
i,j\leq 15)$, $N_{31,j}\leq N_{31,i}$ $(1\leq i,j\leq 64)$,
$N_{31,l},N_{4,j},I_{4,m}\leq N_{4,i}$ $(1\leq i,j\leq 152,1\leq l\leq
64,1\leq m\leq 7)$ and $I_{4,j}\leq I_{4,i}$ $(1\leq i,j\leq 7)$ are given as
in Tables $10,11,12$. In Section 4, we compute the torus invariants
$TI_{G}=[l_{1},l_{2},l_{3},l_{4}]$ of $[M_{G}]^{fl}$ (see Definition 4.3) for
$G=N_{3,i}$, $N_{31,i}$, $N_{4,i}$ and $I_{4,i}$ where $l_{1}=0$ (resp. $1$,
$2$) if $[M_{G}]^{fl}=0$ (resp. $[M_{G}]^{fl}\neq 0$ but invertible,
$[M_{G}]^{fl}$ is not invertible), $l_{2}$, $l_{3}$ and $l_{4}$ are the
abelian invariants of ${}^{1}_{w}(G,[M_{G}]^{fl})$,
${}^{2}_{w}(G,([M_{G}]^{fl})^{\circ})$ and ${}^{2}_{w}(G,[M_{G}]^{fl}$)
respectively. In particular, we see that ${}^{1}_{w}(G,[M_{G}]^{fl})\simeq{\rm
Br}(X)/{\rm Br}(k)$ where $X$ is a smooth $k$-compactification of an algebraic
$k$-torus $T$ with $\widehat{T}=M_{G}$ and ${\rm Br}(X)$ is the Brauer-
Grothendieck group of $X$, see Section 4 for details. In Section 5, by using
the torus invariants $TI_{G}$, we give a partial result for the classification
of the weak stably birationally $k$-equivalent classes of algebraic $k$-tori
$T$ up to dimension $4$. In Section 6, the proofs of Theorem 1.22 and Theorem
1.24 are given. In particular, we get the complete classification of weak
stably $k$-equivalent classes of $[M_{G}]^{fl}$ (resp. $T$) up to dimension
$4$. In Section 7, for $G=N_{3,i}$, $N_{4,i}$ we prove that the $2$-part
$\widetilde{N}_{2}$ $($resp. $3$-part $\widetilde{N}_{3}$$)$ of $[M_{G}]^{fl}$
as a $\mathbbm{Z}_{p}[Sy_{p}(G)]$-lattice (see Definition 7.1) is a faithful
and indecomposable $\mathbbm{Z}_{2}[Sy_{2}(G)]$-lattice $($resp.
$\mathbbm{Z}_{3}[Sy_{3}(G)]$-lattice$)$ unless it vanishes (Tables $14,15$).
In Section 8, the proofs of ${\rm(1)}\Rightarrow{\rm(2)}$ of Theorem 1.23,
Theorem 1.26 and Theorem 1.29 are given. In Section 9, we prove the
implications ${\rm(2)}\Rightarrow{\rm(1)}$ of Theorem 1.23, Theorem 1.26 and
Theorem 1.29 except for the cases $i=j=137,139,145,147$ (resp. $i=j=7$) of
$N_{4,i}$ and $N_{4,j}$ (resp. $I_{4,i}$ and $I_{4,j}$). In Section 10, the
proofs of ${\rm(2)}\Leftrightarrow{\rm(3)}\Leftrightarrow{\rm(4)}$ of Theorem
1.23, Theorem 1.26 and Theorem 1.29 are given. In Section 11, we show Theorem
1.31 which gives relationships among the $I_{4,i}$ cases. In Section 12, we
prove Theorem 1.32 which can answer Problem 1.3 for algebraic $k$-tori $T$ and
$T^{\prime}$ of general dimensions when $T$ (resp. $T^{\prime}$) is stably
birationally $k$-equivalent to some algebraic $k$-torus of dimension up to
$4$.

######  Acknowledgment.

The authors would like to thank Shizuo Endo for giving them useful and
valuable comments related to Krull-Schmidt-Azumaya theorem, in particular,
Lemma 7.11 in Section 7.

##  2\. The groups $N_{3,i}$, $N_{31,i}$, $N_{4,i}$ and $I_{4,i}$: not stably
$k$-rational cases of dimensions $3$ and $4$

We give Tables $6,7,8,9$ for the groups $N_{3,i}$, $N_{31,i}$, $N_{4,i}$ and
$I_{4,i}$ respectively as in Definition 1.18. They are also available as
NonInv.dat from

https://www.math.kyoto-u.ac.jp/~yamasaki/Algorithm/BCAlgtori/.

For $2\leq n\leq 4$, the GAP ID $(n,i,j,k)$ of a finite subgroup $G$ of ${\rm
GL}(n,\mathbbm{Z})$ stands for the $k$-th $\mathbbm{Z}$-class of the $j$-th
$\mathbbm{Q}$-class of the $i$-th crystal system of dimension $n$ in GAP (see
also Hoshi and Yamasaki [HY17, Chapter 3], [BBNWZ78, Table 1]). The SmallGroup
ID $[n,i]$ stands for the $i$-th group of order $n$ in GAP [GAP]. Let
${}^{t}G$ be the group of transposed matrices of $G\leq{\rm
GL}(n,\mathbbm{Z})$. For three groups ${}^{1}_{w}(G,[M_{G}]^{fl})$,
${}^{2}_{w}(G,([M_{G}]^{fl})^{\circ})$, ${}^{2}_{w}(G,[M_{G}]^{fl})$ where
$M_{G}$ be the corresponding $G$-lattice of $\mathbbm{Z}$-rank $n$ as in
Definition 1.21, we will explain and evaluate them in Section 4.

######  Lemma 2.1.

Let $M$ $($resp. $M^{\prime}$$)$ be a $G$-lattice $($resp.
$G^{\prime}$-lattice$)$. Then $M^{\prime\prime}\simeq M\oplus M^{\prime}$ is a
$\widetilde{H}$-lattice where $\widetilde{H}\leq G\times G^{\prime}$ is a
subdirect product of $G$ and $G^{\prime}$ which acts on $M$ $($resp.
$M^{\prime}$$)$ through the surjection $\varphi_{1}:\widetilde{H}\rightarrow
G$ $($resp. $\varphi_{2}:\widetilde{H}\rightarrow G^{\prime}$$)$. If
$[M^{\prime}]^{fl}=0$ as a $G^{\prime}$-lattice, then
$[M^{\prime\prime}]^{fl}\sim[M]^{fl}$.

###### Proof.

Because $[M^{\prime\prime}]^{fl}=[M]^{fl}+[M^{\prime}]^{fl}$ as
$\widetilde{H}$-lattices, we have $[M^{\prime\prime}]^{fl}=[M]^{fl}$ and hence
$[M^{\prime\prime}]^{fl}\sim[M]^{fl}$. ∎

By Lemma 2.1, for $G^{\prime}=N_{31,j}\leq{\rm GL}(4,\mathbbm{Z})$, we can
find $G=N_{3,i}\leq{\rm GL}(3,\mathbbm{Z})$ ($1\leq i\leq 15$) such that
$[M_{G}]^{fl}\sim[M_{G^{\prime}}]^{fl}$ (see Table $7$).

Table $6$: Groups $G=N_{3,i}$ $(1\leq i\leq 15)$ with $L(M)^{G}$ not retract
$k$-rational, rank $M=3$, $M=M_{G}$: indecomposable

$i$ | GAP ID | $N_{3,i}$ | SmallGroup ID |  ${}^{t}G$ in [Kun90] | ${}^{1}_{w}(G,[M]^{fl})$ | ${}^{2}_{w}(G,([M]^{fl})^{\circ})$ | ${}^{2}_{w}(G,[M]^{fl})$
---|---|---|---|---|---|---|---
$1$ | $(3,3,1,3)$ | $C_{2}^{2}$ | $[4,2]$ | $U_{1}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$
$2$ | $(3,3,3,3)$ | $C_{2}^{3}$ | $[8,5]$ | $U_{2}$ | $0$ | $0$ | $0$
$3$ | $(3,3,3,4)$ | $C_{2}^{3}$ | $[8,5]$ | $W_{2}$ | $0$ | $0$ | $0$
$4$ | $(3,4,3,2)$ | $C_{4}\times C_{2}$ | $[8,2]$ | $W_{1}$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$5$ | $(3,4,4,2)$ | $D_{4}$ | $[8,3]$ | $U_{3}$ | $0$ | $0$ | $0$
$6$ | $(3,4,6,3)$ | $D_{4}$ | $[8,3]$ | $U_{4}$ | $0$ | $0$ | $0$
$7$ | $(3,4,7,2)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $U_{6}$ | $0$ | $0$ | $0$
$8$ | $(3,7,1,2)$ | $A_{4}$ | $[12,3]$ | $U_{5}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$
$9$ | $(3,7,2,2)$ | $A_{4}\times C_{2}$ | $[24,13]$ | $U_{7}$ | $0$ | $0$ | $0$
$10$ | $(3,7,2,3)$ | $A_{4}\times C_{2}$ | $[24,13]$ | $W_{3}$ | $0$ | $0$ | $0$
$11$ | $(3,7,3,2)$ | $S_{4}$ | $[24,12]$ | $U_{9}$ | $0$ | $0$ | $0$
$12$ | $(3,7,3,3)$ | $S_{4}$ | $[24,12]$ | $U_{8}$ | $0$ | $0$ | $0$
$13$ | $(3,7,4,2)$ | $S_{4}$ | $[24,12]$ | $U_{10}$ | $0$ | $0$ | $0$
$14$ | $(3,7,5,2)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $U_{12}$ | $0$ | $0$ | $0$
$15$ | $(3,7,5,3)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $U_{11}$ | $0$ | $0$ | $0$

Table $7$: Groups $G^{\prime}=N_{31,j}$ $(1\leq j\leq 64)$ with
$[M_{G^{\prime}}]^{fl}\sim[M_{G}]^{fl}$, $G=N_{3,i}$ $(1\leq i\leq 15)$ and
$L(M)^{G^{\prime}}$ not retract $k$-rational, rank $M=4$,
$M=M_{G^{\prime}}\simeq M_{G}\oplus M_{G^{\prime\prime}}$: decomposable with
rank $M_{G}=3$ and rank $M_{G^{\prime\prime}}=1$

$j$ | GAP ID | $N_{31,j}$ | SmallGroup ID | $i:N_{3,i}$ | $N_{31,j}$ | ${}^{1}_{w}(G,[M]^{fl})$ | ${}^{2}_{w}(G,([M]^{fl})^{\circ})$ | ${}^{2}_{w}(G,[M]^{fl})$
---|---|---|---|---|---|---|---|---
$1$ | $(4,4,3,6)$ | $C_{2}^{2}$ | $[4,2]$ | $1$ |  | $\mathbbm{Z}/2\mathbbm{Z}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$
$2$ | $(4,4,4,4)$ | $C_{2}^{3}$ | $[8,5]$ | $3$ |  | $0$ | $0$ | $0$
$3$ | $(4,4,4,6)$ | $C_{2}^{3}$ | $[8,5]$ | $2$ |  | $0$ | $0$ | $0$
$4$ | $(4,5,1,9)$ | $C_{2}^{2}$ | $[4,2]$ | $1$ | $N_{3,1}\times\\{1\\}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$
$5$ | $(4,5,2,4)$ | $C_{2}^{3}$ | $[8,5]$ | $3$ |  | $0$ | $0$ | $0$
$6$ | $(4,5,2,7)$ | $C_{2}^{3}$ | $[8,5]$ | $2$ |  | $0$ | $0$ | $0$
$7$ | $(4,6,1,4)$ | $C_{2}^{3}$ | $[8,5]$ | $3$ | $N_{3,3}\times\\{1\\}$ | $0$ | $0$ | $0$
$8$ | $(4,6,1,8)$ | $C_{2}^{3}$ | $[8,5]$ | $2$ | $N_{3,2}\times\\{1\\}$ | $0$ | $0$ | $0$
$9$ | $(4,6,2,4)$ | $C_{2}^{3}$ | $[8,5]$ | $3$ |  | $0$ | $0$ | $0$
$10$ | $(4,6,2,8)$ | $C_{2}^{3}$ | $[8,5]$ | $2$ |  | $0$ | $0$ | $0$
$11$ | $(4,6,2,9)$ | $C_{2}^{3}$ | $[8,5]$ | $1$ | $N_{3,1}\times\\{\pm 1\\}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$
$12$ | $(4,6,3,3)$ | $C_{2}^{4}$ | $[16,14]$ | $3$ | $N_{3,3}\times\\{\pm 1\\}$ | $0$ | $0$ | $0$
$13$ | $(4,6,3,6)$ | $C_{2}^{4}$ | $[16,14]$ | $2$ | $N_{3,2}\times\\{\pm 1\\}$ | $0$ | $0$ | $0$
$14$ | $(4,7,3,2)$ | $C_{4}\times C_{2}$ | $[8,2]$ | $4$ |  | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$15$ | $(4,7,4,3)$ | $D_{4}$ | $[8,3]$ | $6$ |  | $0$ | $0$ | $0$
$16$ | $(4,7,5,2)$ | $D_{4}$ | $[8,3]$ | $5$ |  | $0$ | $0$ | $0$
$17$ | $(4,7,7,2)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $7$ |  | $0$ | $0$ | $0$
$18$ | $(4,12,2,4)$ | $C_{4}\times C_{2}$ | $[8,2]$ | $4$ |  | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$19$ | $(4,12,3,7)$ | $D_{4}$ | $[8,3]$ | $6$ | $N_{3,6}\times\\{1\\}$ | $0$ | $0$ | $0$
$20$ | $(4,12,4,6)$ | $D_{4}$ | $[8,3]$ | $6$ |  | $0$ | $0$ | $0$
$21$ | $(4,12,4,8)$ | $D_{4}$ | $[8,3]$ | $5$ |  | $0$ | $0$ | $0$
$22$ | $(4,12,4,9)$ | $D_{4}$ | $[8,3]$ | $5$ |  | $0$ | $0$ | $0$
$23$ | $(4,12,5,6)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $7$ |  | $0$ | $0$ | $0$
$24$ | $(4,12,5,7)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $7$ |  | $0$ | $0$ | $0$
$25$ | $(4,13,1,3)$ | $C_{4}\times C_{2}$ | $[8,2]$ | $4$ | $N_{3,4}\times\\{1\\}$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$26$ | $(4,13,2,4)$ | $C_{4}\times C_{2}$ | $[8,2]$ | $4$ |  | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$27$ | $(4,13,3,4)$ | $D_{4}$ | $[8,3]$ | $6$ |  | $0$ | $0$ | $0$
$28$ | $(4,13,4,3)$ | $D_{4}$ | $[8,3]$ | $5$ | $N_{3,5}\times\\{1\\}$ | $0$ | $0$ | $0$
$29$ | $(4,13,5,3)$ | $C_{4}\times C_{2}^{2}$ | $[16,10]$ | $4$ | $N_{3,4}\times\\{\pm 1\\}$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$30$ | $(4,13,6,3)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $7$ | $N_{3,7}\times\\{1\\}$ | $0$ | $0$ | $0$
$31$ | $(4,13,7,6)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $6$ | $N_{3,6}\times\\{\pm 1\\}$ | $0$ | $0$ | $0$
$32$ | $(4,13,7,7)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $7$ |  | $0$ | $0$ | $0$
$33$ | $(4,13,7,8)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $7$ |  | $0$ | $0$ | $0$
$34$ | $(4,13,8,3)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $7$ |  | $0$ | $0$ | $0$
$35$ | $(4,13,8,4)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $5$ | $N_{3,5}\times\\{\pm 1\\}$ | $0$ | $0$ | $0$
$36$ | $(4,13,9,3)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $7$ |  | $0$ | $0$ | $0$
$37$ | $(4,13,10,3)$ | $D_{4}\times C_{2}^{2}$ | $[32,46]$ | $7$ | $N_{3,7}\times\\{\pm 1\\}$ | $0$ | $0$ | $0$
$38$ | $(4,24,1,5)$ | $A_{4}$ | $[12,3]$ | $8$ | $N_{3,8}\times\\{1\\}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$
$39$ | $(4,24,2,3)$ | $A_{4}\times C_{2}$ | $[24,13]$ | $10$ |  | $0$ | $0$ | $0$
$40$ | $(4,24,2,5)$ | $A_{4}\times C_{2}$ | $[24,13]$ | $9$ |  | $0$ | $0$ | $0$
$41$ | $(4,24,3,5)$ | $S_{4}$ | $[24,12]$ | $13$ | $N_{3,13}\times\\{1\\}$ | $0$ | $0$ | $0$
$42$ | $(4,24,4,3)$ | $S_{4}$ | $[24,12]$ | $12$ |  | $0$ | $0$ | $0$
$43$ | $(4,24,4,5)$ | $S_{4}$ | $[24,12]$ | $11$ |  | $0$ | $0$ | $0$
$44$ | $(4,24,5,3)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $15$ |  | $0$ | $0$ | $0$
$45$ | $(4,24,5,5)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $14$ |  | $0$ | $0$ | $0$
$46$ | $(4,25,1,2)$ | $A_{4}\times C_{2}$ | $[24,13]$ | $10$ | $N_{3,10}\times\\{1\\}$ | $0$ | $0$ | $0$
$47$ | $(4,25,1,4)$ | $A_{4}\times C_{2}$ | $[24,13]$ | $9$ | $N_{3,9}\times\\{1\\}$ | $0$ | $0$ | $0$
$48$ | $(4,25,2,4)$ | $A_{4}\times C_{2}$ | $[24,13]$ | $8$ | $N_{3,8}\times\\{\pm 1\\}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$
$49$ | $(4,25,3,2)$ | $S_{4}$ | $[24,12]$ | $12$ | $N_{3,12}\times\\{1\\}$ | $0$ | $0$ | $0$
$50$ | $(4,25,3,4)$ | $S_{4}$ | $[24,12]$ | $11$ | $N_{3,11}\times\\{1\\}$ | $0$ | $0$ | $0$
$51$ | $(4,25,4,4)$ | $S_{4}$ | $[24,12]$ | $13$ |  | $0$ | $0$ | $0$
$52$ | $(4,25,5,2)$ | $A_{4}\times C_{2}^{2}$ | $[48,49]$ | $10$ | $N_{3,10}\times\\{\pm 1\\}$ | $0$ | $0$ | $0$
$53$ | $(4,25,5,4)$ | $A_{4}\times C_{2}^{2}$ | $[48,49]$ | $9$ | $N_{3,9}\times\\{\pm 1\\}$ | $0$ | $0$ | $0$
$54$ | $(4,25,6,2)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $15$ |  | $0$ | $0$ | $0$
$55$ | $(4,25,6,4)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $14$ |  | $0$ | $0$ | $0$
$56$ | $(4,25,7,2)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $15$ | $N_{3,15}\times\\{1\\}$ | $0$ | $0$ | $0$
$57$ | $(4,25,7,4)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $14$ | $N_{3,14}\times\\{1\\}$ | $0$ | $0$ | $0$
$58$ | $(4,25,8,2)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $15$ |  | $0$ | $0$ | $0$
$59$ | $(4,25,8,4)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $14$ |  | $0$ | $0$ | $0$
$60$ | $(4,25,9,4)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $13$ | $N_{3,13}\times\\{\pm 1\\}$ | $0$ | $0$ | $0$
$61$ | $(4,25,10,2)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $12$ | $N_{3,12}\times\\{\pm 1\\}$ | $0$ | $0$ | $0$
$62$ | $(4,25,10,4)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $11$ | $N_{3,11}\times\\{\pm 1\\}$ | $0$ | $0$ | $0$
$63$ | $(4,25,11,2)$ | $S_{4}\times C_{2}^{2}$ | $[96,226]$ | $15$ | $N_{3,15}\times\\{\pm 1\\}$ | $0$ | $0$ | $0$
$64$ | $(4,25,11,4)$ | $S_{4}\times C_{2}^{2}$ | $[96,226]$ | $14$ | $N_{3,14}\times\\{\pm 1\\}$ | $0$ | $0$ | $0$

Table $8$: Groups $G=N_{4,i}$ $(1\leq i\leq 152)$ with $L(M)^{G}$ not retract
$k$-rational, rank $M=4$, $M=M_{G}$: indecomposable

$i$ | GAP ID | $N_{4,i}$ | SmallGroup ID | ${}^{1}_{w}(G,[M]^{fl})$ | ${}^{2}_{w}(G,([M]^{fl})^{\circ})$ | ${}^{2}_{w}(G,[M]^{fl})$
---|---|---|---|---|---|---
$1$ | $(4,5,1,12)$ | $C_{2}^{2}$ | $[4,2]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$
$2$ | $(4,5,2,5)$ | $C_{2}^{3}$ | $[8,5]$ | $0$ | $0$ | $0$
$3$ | $(4,5,2,8)$ | $C_{2}^{3}$ | $[8,5]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$ | $0$
$4$ | $(4,5,2,9)$ | $C_{2}^{3}$ | $[8,5]$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$5$ | $(4,6,1,6)$ | $C_{2}^{3}$ | $[8,5]$ | $0$ | $0$ | $0$
$6$ | $(4,6,1,11)$ | $C_{2}^{3}$ | $[8,5]$ | $0$ | $0$ | $0$
$7$ | $(4,6,2,6)$ | $C_{2}^{3}$ | $[8,5]$ | $0$ | $0$ | $0$
$8$ | $(4,6,2,10)$ | $C_{2}^{3}$ | $[8,5]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$ | $0$
$9$ | $(4,6,2,12)$ | $C_{2}^{3}$ | $[8,5]$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$
$10$ | $(4,6,3,4)$ | $C_{2}^{4}$ | $[16,14]$ | $0$ | $0$ | $0$
$11$ | $(4,6,3,7)$ | $C_{2}^{4}$ | $[16,14]$ | $0$ | $0$ | $0$
$12$ | $(4,6,3,8)$ | $C_{2}^{4}$ | $[16,14]$ | $0$ | $0$ | $0$
$13$ | $(4,12,2,5)$ | $C_{4}\times C_{2}$ | $[8,2]$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$14$ | $(4,12,2,6)$ | $C_{4}\times C_{2}$ | $[8,2]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$15$ | $(4,12,3,11)$ | $D_{4}$ | $[8,3]$ | $0$ | $0$ | $0$
$16$ | $(4,12,4,10)$ | $D_{4}$ | $[8,3]$ | $0$ | $0$ | $0$
$17$ | $(4,12,4,11)$ | $D_{4}$ | $[8,3]$ | $0$ | $0$ | $0$
$18$ | $(4,12,4,12)$ | $D_{4}$ | $[8,3]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$
$19$ | $(4,12,5,8)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $0$ | $0$ | $0$
$20$ | $(4,12,5,9)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $0$ | $0$ | $0$
$21$ | $(4,12,5,10)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$ | $0$
$22$ | $(4,12,5,11)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $0$ | $0$ | $0$
$23$ | $(4,13,1,5)$ | $C_{4}\times C_{2}$ | $[8,2]$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$24$ | $(4,13,2,5)$ | $C_{4}\times C_{2}$ | $[8,2]$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$25$ | $(4,13,3,5)$ | $D_{4}$ | $[8,3]$ | $0$ | $0$ | $0$
$26$ | $(4,13,4,5)$ | $D_{4}$ | $[8,3]$ | $0$ | $0$ | $0$
$27$ | $(4,13,5,4)$ | $C_{4}\times C_{2}^{2}$ | $[16,10]$ | $0$ | $0$ | $(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$
$28$ | $(4,13,5,5)$ | $C_{4}\times C_{2}^{2}$ | $[16,10]$ | $0$ | $0$ | $0$
$29$ | $(4,13,6,5)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $0$ | $0$ | $0$
$30$ | $(4,13,7,9)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $0$ | $0$ | $0$
$31$ | $(4,13,7,10)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $0$ | $0$ | $0$
$32$ | $(4,13,7,11)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $0$ | $0$ | $0$
$33$ | $(4,13,8,5)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $0$ | $0$ | $0$
$34$ | $(4,13,8,6)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $0$ | $0$ | $0$
$35$ | $(4,13,9,4)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $0$ | $0$ | $0$
$36$ | $(4,13,9,5)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $0$ | $0$ | $0$
$37$ | $(4,13,10,4)$ | $D_{4}\times C_{2}^{2}$ | $[32,46]$ | $0$ | $0$ | $0$
$38$ | $(4,13,10,5)$ | $D_{4}\times C_{2}^{2}$ | $[32,46]$ | $0$ | $0$ | $0$
$39$ | $(4,18,1,3)$ | $C_{4}\times C_{2}$ | $[8,2]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$40$ | $(4,18,2,4)$ | $(C_{4}\times C_{2})\rtimes C_{2}$ | $[16,3]$ | $0$ | $0$ | $0$
$41$ | $(4,18,2,5)$ | $(C_{4}\times C_{2})\rtimes C_{2}$ | $[16,3]$ | $0$ | $0$ | $0$
$42$ | $(4,18,3,5)$ | $(C_{4}\times C_{2})\rtimes C_{2}$ | $[16,3]$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$43$ | $(4,18,3,6)$ | $(C_{4}\times C_{2})\rtimes C_{2}$ | $[16,3]$ | $0$ | $0$ | $0$
$44$ | $(4,18,3,7)$ | $(C_{4}\times C_{2})\rtimes C_{2}$ | $[16,3]$ | $0$ | $0$ | $0$
$45$ | $(4,18,4,4)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$ | $0$
$46$ | $(4,18,4,5)$ | $D_{4}\times C_{2}$ | $[16,11]$ | $0$ | $0$ | $0$
$47$ | $(4,18,5,5)$ | $C_{2}^{4}\rtimes C_{2}$ | $[32,27]$ | $0$ | $0$ | $0$
$48$ | $(4,18,5,6)$ | $C_{2}^{4}\rtimes C_{2}$ | $[32,27]$ | $0$ | $0$ | $0$
$49$ | $(4,18,5,7)$ | $C_{2}^{4}\rtimes C_{2}$ | $[32,27]$ | $0$ | $0$ | $0$
$50$ | $(4,19,1,2)$ | $C_{4}\rtimes C_{4}$ | $[16,4]$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$51$ | $(4,19,2,2)$ | $C_{4}^{2}$ | $[16,2]$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$52$ | $(4,19,3,2)$ | $D_{4}\times C_{4}$ | $[32,25]$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$53$ | $(4,19,4,3)$ | $(C_{4}\times C_{2}^{2})\rtimes C_{2}$ | $[32,28]$ | $0$ | $0$ | $0$
$54$ | $(4,19,4,4)$ | $(C_{4}\times C_{2}^{2})\rtimes C_{2}$ | $[32,28]$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$55$ | $(4,19,5,2)$ | $C_{4}^{2}\rtimes C_{2}$ | $[32,34]$ | $0$ | $0$ | $0$
$56$ | $(4,19,6,2)$ | $D_{4}^{2}$ | $[64,226]$ | $0$ | $0$ | $0$
$57$ | $(4,22,1,1)$ | $C_{3}^{2}$ | $[9,2]$ | $0$ | $0$ | $\mathbbm{Z}/3\mathbbm{Z}$
$58$ | $(4,22,2,1)$ | $C_{6}\times C_{3}$ | $[18,5]$ | $0$ | $0$ | $0$
$59$ | $(4,22,3,1)$ | $S_{3}\times C_{3}$ | $[18,3]$ | $0$ | $0$ | $0$
$60$ | $(4,22,4,1)$ | $S_{3}\times C_{3}$ | $[18,3]$ | $0$ | $0$ | $\mathbbm{Z}/3\mathbbm{Z}$
$61$ | $(4,22,5,1)$ | $C_{3}^{2}\rtimes C_{2}$ | $[18,4]$ | $0$ | $0$ | $\mathbbm{Z}/3\mathbbm{Z}$
$62$ | $(4,22,5,2)$ | $C_{3}^{2}\rtimes C_{2}$ | $[18,4]$ | $0$ | $0$ | $0$
$63$ | $(4,22,6,1)$ | $S_{3}\times C_{6}$ | $[36,12]$ | $0$ | $0$ | $0$
$64$ | $(4,22,7,1)$ | $(C_{3}^{2}\rtimes C_{2})\times C_{2}$ | $[36,13]$ | $0$ | $0$ | $0$
$65$ | $(4,22,8,1)$ | $S_{3}^{2}$ | $[36,10]$ | $0$ | $0$ | $0$
$66$ | $(4,22,9,1)$ | $S_{3}^{2}$ | $[36,10]$ | $0$ | $0$ | $0$
$67$ | $(4,22,10,1)$ | $S_{3}^{2}$ | $[36,10]$ | $0$ | $0$ | $\mathbbm{Z}/3\mathbbm{Z}$
$68$ | $(4,22,11,1)$ | $S_{3}^{2}\times C_{2}$ | $[72,46]$ | $0$ | $0$ | $0$
$69$ | $(4,24,2,4)$ | $A_{4}\times C_{2}$ | $[24,13]$ | $0$ | $0$ | $0$
$70$ | $(4,24,2,6)$ | $A_{4}\times C_{2}$ | $[24,13]$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$71$ | $(4,24,4,4)$ | $S_{4}$ | $[24,12]$ | $0$ | $0$ | $0$
$72$ | $(4,24,5,4)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $0$ | $0$ | $0$
$73$ | $(4,24,5,6)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $0$ | $0$ | $0$
$74$ | $(4,25,1,3)$ | $A_{4}\times C_{2}$ | $[24,13]$ | $0$ | $0$ | $0$
$75$ | $(4,25,2,3)$ | $A_{4}\times C_{2}$ | $[24,13]$ | $0$ | $0$ | $0$
$76$ | $(4,25,2,5)$ | $A_{4}\times C_{2}$ | $[24,13]$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$
$77$ | $(4,25,3,3)$ | $S_{4}$ | $[24,12]$ | $0$ | $0$ | $0$
$78$ | $(4,25,4,3)$ | $S_{4}$ | $[24,12]$ | $0$ | $0$ | $0$
$79$ | $(4,25,5,3)$ | $A_{4}\times C_{2}^{2}$ | $[48,49]$ | $0$ | $0$ | $0$
$80$ | $(4,25,5,5)$ | $A_{4}\times C_{2}^{2}$ | $[48,49]$ | $0$ | $0$ | $0$
$81$ | $(4,25,6,3)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $0$ | $0$ | $0$
$82$ | $(4,25,6,5)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $0$ | $0$ | $0$
$83$ | $(4,25,7,3)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $0$ | $0$ | $0$
$84$ | $(4,25,8,3)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $0$ | $0$ | $0$
$85$ | $(4,25,9,3)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $0$ | $0$ | $0$
$86$ | $(4,25,9,5)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $0$ | $0$ | $0$
$87$ | $(4,25,10,3)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $0$ | $0$ | $0$
$88$ | $(4,25,10,5)$ | $S_{4}\times C_{2}$ | $[48,48]$ | $0$ | $0$ | $0$
$89$ | $(4,25,11,3)$ | $S_{4}\times C_{2}^{2}$ | $[96,226]$ | $0$ | $0$ | $0$
$90$ | $(4,25,11,5)$ | $S_{4}\times C_{2}^{2}$ | $[96,226]$ | $0$ | $0$ | $0$
$91$ | $(4,29,1,1)$ | $S_{3}\times C_{3}$ | $[18,3]$ | $0$ | $0$ | $0$
$92$ | $(4,29,1,2)$ | $S_{3}\times C_{3}$ | $[18,3]$ | $0$ | $0$ | $\mathbbm{Z}/3\mathbbm{Z}$
$93$ | $(4,29,2,1)$ | $S_{3}\times C_{6}$ | $[36,12]$ | $0$ | $0$ | $0$
$94$ | $(4,29,3,1)$ | $S_{3}^{2}$ | $[36,10]$ | $0$ | $0$ | $0$
$95$ | $(4,29,3,2)$ | $S_{3}^{2}$ | $[36,10]$ | $0$ | $0$ | $\mathbbm{Z}/3\mathbbm{Z}$
$96$ | $(4,29,3,3)$ | $S_{3}^{2}$ | $[36,10]$ | $0$ | $0$ | $0$
$97$ | $(4,29,4,1)$ | $C_{3}^{2}\rtimes C_{4}$ | $[36,9]$ | $0$ | $0$ | $\mathbbm{Z}/3\mathbbm{Z}$
$98$ | $(4,29,4,2)$ | $C_{3}^{2}\rtimes C_{4}$ | $[36,9]$ | $0$ | $0$ | $0$
$99$ | $(4,29,5,1)$ | $S_{3}^{2}\times C_{2}$ | $[72,46]$ | $0$ | $0$ | $0$
$100$ | $(4,29,6,1)$ | $(C_{3}^{2}\rtimes C_{4})\times C_{2}$ | $[72,45]$ | $0$ | $0$ | $0$
$101$ | $(4,29,7,1)$ | $S_{3}^{2}\rtimes C_{2}$ | $[72,40]$ | $0$ | $0$ | $0$
$102$ | $(4,29,7,2)$ | $S_{3}^{2}\rtimes C_{2}$ | $[72,40]$ | $0$ | $0$ | $0$
$103$ | $(4,29,8,1)$ | $S_{3}^{2}\rtimes C_{2}$ | $[72,40]$ | $0$ | $0$ | $0$
$104$ | $(4,29,8,2)$ | $S_{3}^{2}\rtimes C_{2}$ | $[72,40]$ | $0$ | $0$ | $\mathbbm{Z}/3\mathbbm{Z}$
$105$ | $(4,29,9,1)$ | $(S_{3}^{2}\rtimes C_{2})\times C_{2}$ | $[144,186]$ | $0$ | $0$ | $0$
$106$ | $(4,32,1,2)$ | $Q_{8}$ | $[8,4]$ | $(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$ | $0$ | $(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$
$107$ | $(4,32,2,2)$ | $C_{8}\rtimes C_{2}$ | $[16,6]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$108$ | $(4,32,3,2)$ | $QD_{8}$ | $[16,8]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$109$ | $(4,32,4,2)$ | $(C_{4}\times C_{2})\rtimes C_{2}$ | $[16,13]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$ | $0$
$110$ | $(4,32,5,2)$ | ${\rm SL}(2,\mathbbm{F}_{3})$ | $[24,3]$ | $0$ | $0$ | $0$
$111$ | $(4,32,5,3)$ | ${\rm SL}(2,\mathbbm{F}_{3})$ | $[24,3]$ | $0$ | $0$ | $(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$
$112$ | $(4,32,6,2)$ | $C_{8}\rtimes C_{2}^{2}$ | $[32,43]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$ | $0$
$113$ | $(4,32,7,2)$ | $(C_{8}\rtimes C_{2})\rtimes C_{2}$ | $[32,7]$ | $0$ | $0$ | $0$
$114$ | $(4,32,8,2)$ | $C_{4}^{2}\rtimes C_{2}$ | $[32,11]$ | $0$ | $0$ | $0$
$115$ | $(4,32,9,4)$ | $C_{2}^{3}\rtimes C_{4}$ | $[32,6]$ | $0$ | $0$ | $0$
$116$ | $(4,32,9,5)$ | $C_{2}^{3}\rtimes C_{4}$ | $[32,6]$ | $0$ | $0$ | $0$
$117$ | $(4,32,10,2)$ | $C_{2}^{3}\rtimes C_{2}^{2}$ | $[32,49]$ | $0$ | $0$ | $0$
$118$ | $(4,32,11,2)$ | ${\rm GL}(2,\mathbbm{F}_{3})$ | $[48,29]$ | $0$ | $0$ | $0$
$119$ | $(4,32,11,3)$ | ${\rm GL}(2,\mathbbm{F}_{3})$ | $[48,29]$ | $0$ | $0$ | $\mathbbm{Z}/2\mathbbm{Z}$
$120$ | $(4,32,12,2)$ | $((C_{8}\rtimes C_{2})\rtimes C_{2})\rtimes C_{2}$ | $[64,134]$ | $0$ | $0$ | $0$
$121$ | $(4,32,13,3)$ | $((C_{8}\rtimes C_{2})\rtimes C_{2})\rtimes C_{2}$ | $[64,32]$ | $0$ | $0$ | $0$
$122$ | $(4,32,13,4)$ | $((C_{8}\rtimes C_{2})\rtimes C_{2})\rtimes C_{2}$ | $[64,32]$ | $0$ | $0$ | $0$
$123$ | $(4,32,14,3)$ | $(C_{2}^{4}\rtimes C_{2})\rtimes C_{2}$ | $[64,138]$ | $0$ | $0$ | $0$
$124$ | $(4,32,14,4)$ | $(C_{2}^{4}\rtimes C_{2})\rtimes C_{2}$ | $[64,138]$ | $0$ | $0$ | $0$
$125$ | $(4,32,15,2)$ | $(C_{2}^{3}\rtimes C_{4})\rtimes C_{2}$ | $[64,34]$ | $0$ | $0$ | $0$
$126$ | $(4,32,16,2)$ | $(C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3}$ | $[96,204]$ | $0$ | $0$ | $0$
$127$ | $(4,32,16,3)$ | $(C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3}$ | $[96,204]$ | $0$ | $0$ | $0$
$128$ | $(4,32,17,2)$ | $D_{4}^{2}\rtimes C_{2}$ | $[128,928]$ | $0$ | $0$ | $0$
$129$ | $(4,32,18,2)$ | $((C_{2}^{4}\rtimes C_{2})\rtimes C_{2})\rtimes C_{3}$ | $[192,201]$ | $0$ | $0$ | $0$
$130$ | $(4,32,18,3)$ | $((C_{2}^{4}\rtimes C_{2})\rtimes C_{2})\rtimes C_{3}$ | $[192,201]$ | $0$ | $0$ | $0$
$131$ | $(4,32,19,2)$ | $((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2}$ | $[192,1493]$ | $0$ | $0$ | $0$
$132$ | $(4,32,19,3)$ | $((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2}$ | $[192,1493]$ | $0$ | $0$ | $0$
$133$ | $(4,32,20,2)$ | $((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2}$ | $[192,1494]$ | $0$ | $0$ | $0$
$134$ | $(4,32,20,3)$ | $((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2}$ | $[192,1494]$ | $0$ | $0$ | $0$
$135$ | $(4,32,21,2)$ | $(((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2})\rtimes C_{2}$ | $[384,5602]$ | $0$ | $0$ | $0$
$136$ | $(4,32,21,3)$ | $(((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2})\rtimes C_{2}$ | $[384,5602]$ | $0$ | $0$ | $0$
$137$ | $(4,33,1,1)$ | $Q_{8}\times C_{3}$ | $[24,11]$ | $0$ | $0$ | $0$
$138$ | $(4,33,3,1)$ | ${\rm SL}(2,\mathbbm{F}_{3})$ | $[24,3]$ | $(\mathbbm{Z}/2\mathbbm{Z})^{\oplus 2}$ | $0$ | $0$
$139$ | $(4,33,4,1)$ | $(Q_{8}\times C_{3})\rtimes C_{2}$ | $[48,17]$ | $0$ | $0$ | $0$
$140$ | $(4,33,5,1)$ | $((C_{4}\times C_{2})\rtimes C_{2})\rtimes C_{3}$ | $[48,33]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$ | $0$
$141$ | $(4,33,6,1)$ | ${\rm GL}(2,\mathbbm{F}_{3})$ | $[48,29]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$ | $0$
$142$ | $(4,33,7,1)$ | ${\rm SL}(2,\mathbbm{F}_{3})\times C_{3}$ | $[72,25]$ | $0$ | $0$ | $0$
$143$ | $(4,33,8,1)$ | $(C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3}$ | $[96,201]$ | $0$ | $0$ | $0$
$144$ | $(4,33,9,1)$ | ${\rm GL}(2,\mathbbm{F}_{3})\rtimes C_{2}$ | $[96,193]$ | $\mathbbm{Z}/2\mathbbm{Z}$ | $0$ | $0$
$145$ | $(4,33,10,1)$ | ${\rm SL}(2,\mathbbm{F}_{3})\rtimes C_{4}$ | $[96,67]$ | $0$ | $0$ | $0$
$146$ | $(4,33,11,1)$ | $({\rm SL}(2,\mathbbm{F}_{3})\times C_{3})\rtimes C_{2}$ | $[144,125]$ | $0$ | $0$ | $0$
$147$ | $(4,33,12,1)$ | $({\rm GL}(2,\mathbbm{F}_{3})\rtimes C_{2})\rtimes C_{2}$ | $[192,988]$ | $0$ | $0$ | $0$
$148$ | $(4,33,13,1)$ | $(C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3}^{2}$ | $[288,860]$ | $0$ | $0$ | $0$
$149$ | $(4,33,14,1)$ | $(((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2})\rtimes C_{3}$ | $[576,8277]$ | $0$ | $0$ | $0$
$150$ | $(4,33,14,2)$ | $(((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3})\rtimes C_{2})\rtimes C_{3}$ | $[576,8277]$ | $0$ | $0$ | $0$
$151$ | $(4,33,15,1)$ | $((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3}^{2})\rtimes C_{2}$ | $[576,8282]$ | $0$ | $0$ | $0$
$152$ | $(4,33,16,1)$ | $(((C_{2}^{3}\rtimes C_{2}^{2})\rtimes C_{3}^{2})\rtimes C_{2})\rtimes C_{2}$ | $[1152,157478]$ | $0$ | $0$ | $0$

Table $9$: Groups $G=I_{4,i}$ $(1\leq i\leq 7)$ with $L(M)^{G}$ not stably
$k$-rational but retract $k$-rational, rank $M=4$, $M=M_{G}$: indecomposable

$i$ | GAP ID | $I_{4,i}$ | SmallGroup ID | ${}^{1}_{w}(G,[M]^{fl})$ | ${}^{2}_{w}(G,([M]^{fl})^{\circ})$ | ${}^{2}_{w}(G,[M]^{fl})$
---|---|---|---|---|---|---
$1$ | $(4,31,1,3)$ | $F_{20}$ | $[20,3]$ | $0$ | $0$ | $0$
$2$ | $(4,31,1,4)$ | $F_{20}$ | $[20,3]$ | $0$ | $0$ | $0$
$3$ | $(4,31,2,2)$ | $F_{20}\times C_{2}$ | $[40,12]$ | $0$ | $0$ | $0$
$4$ | $(4,31,4,2)$ | $S_{5}$ | $[120,34]$ | $0$ | $0$ | $0$
$5$ | $(4,31,5,2)$ | $S_{5}$ | $[120,34]$ | $0$ | $0$ | $0$
$6$ | $(4,31,7,2)$ | $S_{5}\times C_{2}$ | $[240,189]$ | $0$ | $0$ | $0$
$7$ | $(4,33,2,1)$ | $C_{3}\rtimes C_{8}$ | $[24,1]$ | $0$ | $0$ | $0$

Tables $6,7,8,9$ can be obtained in GAP as follows:

######  Example 2.2 (The groups $N_{3,i}$ as in Table $6$).

    gap> Read("BCAlgTori.gap");
    gap> N3table:=List(N3,x->[Position(N3,x),x,
    > StructureDescription(MatGroupZClass(x[1],x[2],x[3],x[4])),
    > IdSmallGroup(MatGroupZClass(x[1],x[2],x[3],x[4]))]);
    [ [ 1, [ 3, 3, 1, 3 ], "C2 x C2", [ 4, 2 ] ],
      [ 2, [ 3, 3, 3, 3 ], "C2 x C2 x C2", [ 8, 5 ] ],
      [ 3, [ 3, 3, 3, 4 ], "C2 x C2 x C2", [ 8, 5 ] ],
      [ 4, [ 3, 4, 3, 2 ], "C4 x C2", [ 8, 2 ] ],
      [ 5, [ 3, 4, 4, 2 ], "D8", [ 8, 3 ] ],
      [ 6, [ 3, 4, 6, 3 ], "D8", [ 8, 3 ] ],
      [ 7, [ 3, 4, 7, 2 ], "C2 x D8", [ 16, 11 ] ],
      [ 8, [ 3, 7, 1, 2 ], "A4", [ 12, 3 ] ],
      [ 9, [ 3, 7, 2, 2 ], "C2 x A4", [ 24, 13 ] ],
      [ 10, [ 3, 7, 2, 3 ], "C2 x A4", [ 24, 13 ] ],
      [ 11, [ 3, 7, 3, 2 ], "S4", [ 24, 12 ] ],
      [ 12, [ 3, 7, 3, 3 ], "S4", [ 24, 12 ] ],
      [ 13, [ 3, 7, 4, 2 ], "S4", [ 24, 12 ] ],
      [ 14, [ 3, 7, 5, 2 ], "C2 x S4", [ 48, 48 ] ],
      [ 15, [ 3, 7, 5, 3 ], "C2 x S4", [ 48, 48 ] ] ]

######  Example 2.3 (The groups $N_{31,i}$ as in Table $7$).

    gap> Read("BCAlgTori.gap");
    gap> N31table:=List(N31,x->[Position(N31,x),x,
    > StructureDescription(MatGroupZClass(x[1],x[2],x[3],x[4])),
    > IdSmallGroup(MatGroupZClass(x[1],x[2],x[3],x[4])),0,0]);;
    gap> for x in N3 do
    > i:=Position(N3,x);
    > N3M1:=AllSubdirectProducts(MatGroupZClass(x[1],x[2],x[3],x[4]),Group([[-1]]));
    > for G in N3M1 do
    > j:=Position(N31,CrystCatZClass(SubdirectProduct2Matrix(G)));
    > if N31table[j][5]<>0 then Print("error"); fi;
    > N31table[j][5]:=i;
    > if Order(G)=Order(MatGroupZClass(x[1],x[2],x[3],x[4])) then N31table[j][6]:=-1;
    > else N31table[j][6]:=2; fi;
    > od;od;
    gap> for x in N3 do
    > i:=Position(N3,x);
    > N3Z:=AllSubdirectProducts(MatGroupZClass(x[1],x[2],x[3],x[4]),Group([[1]]));
    > for G in N3Z do
    > j:=Position(N31,CrystCatZClass(SubdirectProduct2Matrix(G)));
    > if N31table[j][5]<>0 then Print("error"); fi;
    > N31table[j][5]:=i;
    > N31table[j][6]:=1;
    > od;od;
    gap> N31table;
    [ [ 1, [ 4, 4, 3, 6 ], "C2 x C2", [ 4, 2 ], 1, -1 ],
      [ 2, [ 4, 4, 4, 4 ], "C2 x C2 x C2", [ 8, 5 ], 3, -1 ],
      [ 3, [ 4, 4, 4, 6 ], "C2 x C2 x C2", [ 8, 5 ], 2, -1 ],
      [ 4, [ 4, 5, 1, 9 ], "C2 x C2", [ 4, 2 ], 1, 1 ],
      [ 5, [ 4, 5, 2, 4 ], "C2 x C2 x C2", [ 8, 5 ], 3, -1 ],
      [ 6, [ 4, 5, 2, 7 ], "C2 x C2 x C2", [ 8, 5 ], 2, -1 ],
      [ 7, [ 4, 6, 1, 4 ], "C2 x C2 x C2", [ 8, 5 ], 3, 1 ],
      [ 8, [ 4, 6, 1, 8 ], "C2 x C2 x C2", [ 8, 5 ], 2, 1 ],
      [ 9, [ 4, 6, 2, 4 ], "C2 x C2 x C2", [ 8, 5 ], 3, -1 ],
      [ 10, [ 4, 6, 2, 8 ], "C2 x C2 x C2", [ 8, 5 ], 2, -1 ],
      [ 11, [ 4, 6, 2, 9 ], "C2 x C2 x C2", [ 8, 5 ], 1, 2 ],
      [ 12, [ 4, 6, 3, 3 ], "C2 x C2 x C2 x C2", [ 16, 14 ], 3, 2 ],
      [ 13, [ 4, 6, 3, 6 ], "C2 x C2 x C2 x C2", [ 16, 14 ], 2, 2 ],
      [ 14, [ 4, 7, 3, 2 ], "C4 x C2", [ 8, 2 ], 4, -1 ],
      [ 15, [ 4, 7, 4, 3 ], "D8", [ 8, 3 ], 6, -1 ],
      [ 16, [ 4, 7, 5, 2 ], "D8", [ 8, 3 ], 5, -1 ],
      [ 17, [ 4, 7, 7, 2 ], "C2 x D8", [ 16, 11 ], 7, -1 ],
      [ 18, [ 4, 12, 2, 4 ], "C4 x C2", [ 8, 2 ], 4, -1 ],
      [ 19, [ 4, 12, 3, 7 ], "D8", [ 8, 3 ], 6, 1 ],
      [ 20, [ 4, 12, 4, 6 ], "D8", [ 8, 3 ], 6, -1 ],
      [ 21, [ 4, 12, 4, 8 ], "D8", [ 8, 3 ], 5, -1 ],
      [ 22, [ 4, 12, 4, 9 ], "D8", [ 8, 3 ], 5, -1 ],
      [ 23, [ 4, 12, 5, 6 ], "C2 x D8", [ 16, 11 ], 7, -1 ],
      [ 24, [ 4, 12, 5, 7 ], "C2 x D8", [ 16, 11 ], 7, -1 ],
      [ 25, [ 4, 13, 1, 3 ], "C4 x C2", [ 8, 2 ], 4, 1 ],
      [ 26, [ 4, 13, 2, 4 ], "C4 x C2", [ 8, 2 ], 4, -1 ],
      [ 27, [ 4, 13, 3, 4 ], "D8", [ 8, 3 ], 6, -1 ],
      [ 28, [ 4, 13, 4, 3 ], "D8", [ 8, 3 ], 5, 1 ],
      [ 29, [ 4, 13, 5, 3 ], "C4 x C2 x C2", [ 16, 10 ], 4, 2 ],
      [ 30, [ 4, 13, 6, 3 ], "C2 x D8", [ 16, 11 ], 7, 1 ],
      [ 31, [ 4, 13, 7, 6 ], "C2 x D8", [ 16, 11 ], 6, 2 ],
      [ 32, [ 4, 13, 7, 7 ], "C2 x D8", [ 16, 11 ], 7, -1 ],
      [ 33, [ 4, 13, 7, 8 ], "C2 x D8", [ 16, 11 ], 7, -1 ],
      [ 34, [ 4, 13, 8, 3 ], "C2 x D8", [ 16, 11 ], 7, -1 ],
      [ 35, [ 4, 13, 8, 4 ], "C2 x D8", [ 16, 11 ], 5, 2 ],
      [ 36, [ 4, 13, 9, 3 ], "C2 x D8", [ 16, 11 ], 7, -1 ],
      [ 37, [ 4, 13, 10, 3 ], "C2 x C2 x D8", [ 32, 46 ], 7, 2 ],
      [ 38, [ 4, 24, 1, 5 ], "A4", [ 12, 3 ], 8, 1 ],
      [ 39, [ 4, 24, 2, 3 ], "C2 x A4", [ 24, 13 ], 10, -1 ],
      [ 40, [ 4, 24, 2, 5 ], "C2 x A4", [ 24, 13 ], 9, -1 ],
      [ 41, [ 4, 24, 3, 5 ], "S4", [ 24, 12 ], 13, 1 ],
      [ 42, [ 4, 24, 4, 3 ], "S4", [ 24, 12 ], 12, -1 ],
      [ 43, [ 4, 24, 4, 5 ], "S4", [ 24, 12 ], 11, -1 ],
      [ 44, [ 4, 24, 5, 3 ], "C2 x S4", [ 48, 48 ], 15, -1 ],
      [ 45, [ 4, 24, 5, 5 ], "C2 x S4", [ 48, 48 ], 14, -1 ],
      [ 46, [ 4, 25, 1, 2 ], "C2 x A4", [ 24, 13 ], 10, 1 ],
      [ 47, [ 4, 25, 1, 4 ], "C2 x A4", [ 24, 13 ], 9, 1 ],
      [ 48, [ 4, 25, 2, 4 ], "C2 x A4", [ 24, 13 ], 8, 2 ],
      [ 49, [ 4, 25, 3, 2 ], "S4", [ 24, 12 ], 12, 1 ],
      [ 50, [ 4, 25, 3, 4 ], "S4", [ 24, 12 ], 11, 1 ],
      [ 51, [ 4, 25, 4, 4 ], "S4", [ 24, 12 ], 13, -1 ],
      [ 52, [ 4, 25, 5, 2 ], "C2 x C2 x A4", [ 48, 49 ], 10, 2 ],
      [ 53, [ 4, 25, 5, 4 ], "C2 x C2 x A4", [ 48, 49 ], 9, 2 ],
      [ 54, [ 4, 25, 6, 2 ], "C2 x S4", [ 48, 48 ], 15, -1 ],
      [ 55, [ 4, 25, 6, 4 ], "C2 x S4", [ 48, 48 ], 14, -1 ],
      [ 56, [ 4, 25, 7, 2 ], "C2 x S4", [ 48, 48 ], 15, 1 ],
      [ 57, [ 4, 25, 7, 4 ], "C2 x S4", [ 48, 48 ], 14, 1 ],
      [ 58, [ 4, 25, 8, 2 ], "C2 x S4", [ 48, 48 ], 15, -1 ],
      [ 59, [ 4, 25, 8, 4 ], "C2 x S4", [ 48, 48 ], 14, -1 ],
      [ 60, [ 4, 25, 9, 4 ], "C2 x S4", [ 48, 48 ], 13, 2 ],
      [ 61, [ 4, 25, 10, 2 ], "C2 x S4", [ 48, 48 ], 12, 2 ],
      [ 62, [ 4, 25, 10, 4 ], "C2 x S4", [ 48, 48 ], 11, 2 ],
      [ 63, [ 4, 25, 11, 2 ], "C2 x C2 x S4", [ 96, 226 ], 15, 2 ],
      [ 64, [ 4, 25, 11, 4 ], "C2 x C2 x S4", [ 96, 226 ], 14, 2 ] ]

######  Example 2.4 (The groups $N_{4,i}$ as in Table $8$).

    gap> Read("BCAlgTori.gap");
    gap> N4table:=List(N4,x->[Position(N4,x),x,
    > StructureDescription(MatGroupZClass(x[1],x[2],x[3],x[4])),
    > IdSmallGroup(MatGroupZClass(x[1],x[2],x[3],x[4]))]);
    [ [ 1, [ 4, 5, 1, 12 ], "C2 x C2", [ 4, 2 ] ],
      [ 2, [ 4, 5, 2, 5 ], "C2 x C2 x C2", [ 8, 5 ] ],
      [ 3, [ 4, 5, 2, 8 ], "C2 x C2 x C2", [ 8, 5 ] ],
      [ 4, [ 4, 5, 2, 9 ], "C2 x C2 x C2", [ 8, 5 ] ],
      [ 5, [ 4, 6, 1, 6 ], "C2 x C2 x C2", [ 8, 5 ] ],
      [ 6, [ 4, 6, 1, 11 ], "C2 x C2 x C2", [ 8, 5 ] ],
      [ 7, [ 4, 6, 2, 6 ], "C2 x C2 x C2", [ 8, 5 ] ],
      [ 8, [ 4, 6, 2, 10 ], "C2 x C2 x C2", [ 8, 5 ] ],
      [ 9, [ 4, 6, 2, 12 ], "C2 x C2 x C2", [ 8, 5 ] ],
      [ 10, [ 4, 6, 3, 4 ], "C2 x C2 x C2 x C2", [ 16, 14 ] ],
      [ 11, [ 4, 6, 3, 7 ], "C2 x C2 x C2 x C2", [ 16, 14 ] ],
      [ 12, [ 4, 6, 3, 8 ], "C2 x C2 x C2 x C2", [ 16, 14 ] ],
      [ 13, [ 4, 12, 2, 5 ], "C4 x C2", [ 8, 2 ] ],
      [ 14, [ 4, 12, 2, 6 ], "C4 x C2", [ 8, 2 ] ],
      [ 15, [ 4, 12, 3, 11 ], "D8", [ 8, 3 ] ],
      [ 16, [ 4, 12, 4, 10 ], "D8", [ 8, 3 ] ],
      [ 17, [ 4, 12, 4, 11 ], "D8", [ 8, 3 ] ],
      [ 18, [ 4, 12, 4, 12 ], "D8", [ 8, 3 ] ],
      [ 19, [ 4, 12, 5, 8 ], "C2 x D8", [ 16, 11 ] ],
      [ 20, [ 4, 12, 5, 9 ], "C2 x D8", [ 16, 11 ] ],
      [ 21, [ 4, 12, 5, 10 ], "C2 x D8", [ 16, 11 ] ],
      [ 22, [ 4, 12, 5, 11 ], "C2 x D8", [ 16, 11 ] ],
      [ 23, [ 4, 13, 1, 5 ], "C4 x C2", [ 8, 2 ] ],
      [ 24, [ 4, 13, 2, 5 ], "C4 x C2", [ 8, 2 ] ],
      [ 25, [ 4, 13, 3, 5 ], "D8", [ 8, 3 ] ],
      [ 26, [ 4, 13, 4, 5 ], "D8", [ 8, 3 ] ],
      [ 27, [ 4, 13, 5, 4 ], "C4 x C2 x C2", [ 16, 10 ] ],
      [ 28, [ 4, 13, 5, 5 ], "C4 x C2 x C2", [ 16, 10 ] ],
      [ 29, [ 4, 13, 6, 5 ], "C2 x D8", [ 16, 11 ] ],
      [ 30, [ 4, 13, 7, 9 ], "C2 x D8", [ 16, 11 ] ],
      [ 31, [ 4, 13, 7, 10 ], "C2 x D8", [ 16, 11 ] ],
      [ 32, [ 4, 13, 7, 11 ], "C2 x D8", [ 16, 11 ] ],
      [ 33, [ 4, 13, 8, 5 ], "C2 x D8", [ 16, 11 ] ],
      [ 34, [ 4, 13, 8, 6 ], "C2 x D8", [ 16, 11 ] ],
      [ 35, [ 4, 13, 9, 4 ], "C2 x D8", [ 16, 11 ] ],
      [ 36, [ 4, 13, 9, 5 ], "C2 x D8", [ 16, 11 ] ],
      [ 37, [ 4, 13, 10, 4 ], "C2 x C2 x D8", [ 32, 46 ] ],
      [ 38, [ 4, 13, 10, 5 ], "C2 x C2 x D8", [ 32, 46 ] ],
      [ 39, [ 4, 18, 1, 3 ], "C4 x C2", [ 8, 2 ] ],
      [ 40, [ 4, 18, 2, 4 ], "(C4 x C2) : C2", [ 16, 3 ] ],
      [ 41, [ 4, 18, 2, 5 ], "(C4 x C2) : C2", [ 16, 3 ] ],
      [ 42, [ 4, 18, 3, 5 ], "(C4 x C2) : C2", [ 16, 3 ] ],
      [ 43, [ 4, 18, 3, 6 ], "(C4 x C2) : C2", [ 16, 3 ] ],
      [ 44, [ 4, 18, 3, 7 ], "(C4 x C2) : C2", [ 16, 3 ] ],
      [ 45, [ 4, 18, 4, 4 ], "C2 x D8", [ 16, 11 ] ],
      [ 46, [ 4, 18, 4, 5 ], "C2 x D8", [ 16, 11 ] ],
      [ 47, [ 4, 18, 5, 5 ], "(C2 x C2 x C2 x C2) : C2", [ 32, 27 ] ],
      [ 48, [ 4, 18, 5, 6 ], "(C2 x C2 x C2 x C2) : C2", [ 32, 27 ] ],
      [ 49, [ 4, 18, 5, 7 ], "(C2 x C2 x C2 x C2) : C2", [ 32, 27 ] ],
      [ 50, [ 4, 19, 1, 2 ], "C4 : C4", [ 16, 4 ] ],
      [ 51, [ 4, 19, 2, 2 ], "C4 x C4", [ 16, 2 ] ],
      [ 52, [ 4, 19, 3, 2 ], "C4 x D8", [ 32, 25 ] ],
      [ 53, [ 4, 19, 4, 3 ], "(C4 x C2 x C2) : C2", [ 32, 28 ] ],
      [ 54, [ 4, 19, 4, 4 ], "(C4 x C2 x C2) : C2", [ 32, 28 ] ],
      [ 55, [ 4, 19, 5, 2 ], "(C4 x C4) : C2", [ 32, 34 ] ],
      [ 56, [ 4, 19, 6, 2 ], "D8 x D8", [ 64, 226 ] ],
      [ 57, [ 4, 22, 1, 1 ], "C3 x C3", [ 9, 2 ] ],
      [ 58, [ 4, 22, 2, 1 ], "C6 x C3", [ 18, 5 ] ],
      [ 59, [ 4, 22, 3, 1 ], "C3 x S3", [ 18, 3 ] ],
      [ 60, [ 4, 22, 4, 1 ], "C3 x S3", [ 18, 3 ] ],
      [ 61, [ 4, 22, 5, 1 ], "(C3 x C3) : C2", [ 18, 4 ] ],
      [ 62, [ 4, 22, 5, 2 ], "(C3 x C3) : C2", [ 18, 4 ] ],
      [ 63, [ 4, 22, 6, 1 ], "C6 x S3", [ 36, 12 ] ],
      [ 64, [ 4, 22, 7, 1 ], "C2 x ((C3 x C3) : C2)", [ 36, 13 ] ],
      [ 65, [ 4, 22, 8, 1 ], "S3 x S3", [ 36, 10 ] ],
      [ 66, [ 4, 22, 9, 1 ], "S3 x S3", [ 36, 10 ] ],
      [ 67, [ 4, 22, 10, 1 ], "S3 x S3", [ 36, 10 ] ],
      [ 68, [ 4, 22, 11, 1 ], "C2 x S3 x S3", [ 72, 46 ] ],
      [ 69, [ 4, 24, 2, 4 ], "C2 x A4", [ 24, 13 ] ],
      [ 70, [ 4, 24, 2, 6 ], "C2 x A4", [ 24, 13 ] ],
      [ 71, [ 4, 24, 4, 4 ], "S4", [ 24, 12 ] ],
      [ 72, [ 4, 24, 5, 4 ], "C2 x S4", [ 48, 48 ] ],
      [ 73, [ 4, 24, 5, 6 ], "C2 x S4", [ 48, 48 ] ],
      [ 74, [ 4, 25, 1, 3 ], "C2 x A4", [ 24, 13 ] ],
      [ 75, [ 4, 25, 2, 3 ], "C2 x A4", [ 24, 13 ] ],
      [ 76, [ 4, 25, 2, 5 ], "C2 x A4", [ 24, 13 ] ],
      [ 77, [ 4, 25, 3, 3 ], "S4", [ 24, 12 ] ],
      [ 78, [ 4, 25, 4, 3 ], "S4", [ 24, 12 ] ],
      [ 79, [ 4, 25, 5, 3 ], "C2 x C2 x A4", [ 48, 49 ] ],
      [ 80, [ 4, 25, 5, 5 ], "C2 x C2 x A4", [ 48, 49 ] ],
      [ 81, [ 4, 25, 6, 3 ], "C2 x S4", [ 48, 48 ] ],
      [ 82, [ 4, 25, 6, 5 ], "C2 x S4", [ 48, 48 ] ],
      [ 83, [ 4, 25, 7, 3 ], "C2 x S4", [ 48, 48 ] ],
      [ 84, [ 4, 25, 8, 3 ], "C2 x S4", [ 48, 48 ] ],
      [ 85, [ 4, 25, 9, 3 ], "C2 x S4", [ 48, 48 ] ],
      [ 86, [ 4, 25, 9, 5 ], "C2 x S4", [ 48, 48 ] ],
      [ 87, [ 4, 25, 10, 3 ], "C2 x S4", [ 48, 48 ] ],
      [ 88, [ 4, 25, 10, 5 ], "C2 x S4", [ 48, 48 ] ],
      [ 89, [ 4, 25, 11, 3 ], "C2 x C2 x S4", [ 96, 226 ] ],
      [ 90, [ 4, 25, 11, 5 ], "C2 x C2 x S4", [ 96, 226 ] ],
      [ 91, [ 4, 29, 1, 1 ], "C3 x S3", [ 18, 3 ] ],
      [ 92, [ 4, 29, 1, 2 ], "C3 x S3", [ 18, 3 ] ],
      [ 93, [ 4, 29, 2, 1 ], "C6 x S3", [ 36, 12 ] ],
      [ 94, [ 4, 29, 3, 1 ], "S3 x S3", [ 36, 10 ] ],
      [ 95, [ 4, 29, 3, 2 ], "S3 x S3", [ 36, 10 ] ],
      [ 96, [ 4, 29, 3, 3 ], "S3 x S3", [ 36, 10 ] ],
      [ 97, [ 4, 29, 4, 1 ], "(C3 x C3) : C4", [ 36, 9 ] ],
      [ 98, [ 4, 29, 4, 2 ], "(C3 x C3) : C4", [ 36, 9 ] ],
      [ 99, [ 4, 29, 5, 1 ], "C2 x S3 x S3", [ 72, 46 ] ],
      [ 100, [ 4, 29, 6, 1 ], "C2 x ((C3 x C3) : C4)", [ 72, 45 ] ],
      [ 101, [ 4, 29, 7, 1 ], "(S3 x S3) : C2", [ 72, 40 ] ],
      [ 102, [ 4, 29, 7, 2 ], "(S3 x S3) : C2", [ 72, 40 ] ],
      [ 103, [ 4, 29, 8, 1 ], "(S3 x S3) : C2", [ 72, 40 ] ],
      [ 104, [ 4, 29, 8, 2 ], "(S3 x S3) : C2", [ 72, 40 ] ],
      [ 105, [ 4, 29, 9, 1 ], "C2 x ((S3 x S3) : C2)", [ 144, 186 ] ],
      [ 106, [ 4, 32, 1, 2 ], "Q8", [ 8, 4 ] ],
      [ 107, [ 4, 32, 2, 2 ], "C8 : C2", [ 16, 6 ] ],
      [ 108, [ 4, 32, 3, 2 ], "QD16", [ 16, 8 ] ],
      [ 109, [ 4, 32, 4, 2 ], "(C4 x C2) : C2", [ 16, 13 ] ],
      [ 110, [ 4, 32, 5, 2 ], "SL(2,3)", [ 24, 3 ] ],
      [ 111, [ 4, 32, 5, 3 ], "SL(2,3)", [ 24, 3 ] ],
      [ 112, [ 4, 32, 6, 2 ], "C8 : (C2 x C2)", [ 32, 43 ] ],
      [ 113, [ 4, 32, 7, 2 ], "(C8 : C2) : C2", [ 32, 7 ] ],
      [ 114, [ 4, 32, 8, 2 ], "(C4 x C4) : C2", [ 32, 11 ] ],
      [ 115, [ 4, 32, 9, 4 ], "(C2 x C2 x C2) : C4", [ 32, 6 ] ],
      [ 116, [ 4, 32, 9, 5 ], "(C2 x C2 x C2) : C4", [ 32, 6 ] ],
      [ 117, [ 4, 32, 10, 2 ], "(C2 x C2 x C2) : (C2 x C2)", [ 32, 49 ] ],
      [ 118, [ 4, 32, 11, 2 ], "GL(2,3)", [ 48, 29 ] ],
      [ 119, [ 4, 32, 11, 3 ], "GL(2,3)", [ 48, 29 ] ],
      [ 120, [ 4, 32, 12, 2 ], "((C8 : C2) : C2) : C2", [ 64, 134 ] ],
      [ 121, [ 4, 32, 13, 3 ], "((C8 : C2) : C2) : C2", [ 64, 32 ] ],
      [ 122, [ 4, 32, 13, 4 ], "((C8 : C2) : C2) : C2", [ 64, 32 ] ],
      [ 123, [ 4, 32, 14, 3 ], "((C2 x C2 x C2 x C2) : C2) : C2", [ 64, 138 ] ],
      [ 124, [ 4, 32, 14, 4 ], "((C2 x C2 x C2 x C2) : C2) : C2", [ 64, 138 ] ],
      [ 125, [ 4, 32, 15, 2 ], "((C2 x C2 x C2) : C4) : C2", [ 64, 34 ] ],
      [ 126, [ 4, 32, 16, 2 ], "((C2 x C2 x C2) : (C2 x C2)) : C3", [ 96, 204 ] ],
      [ 127, [ 4, 32, 16, 3 ], "((C2 x C2 x C2) : (C2 x C2)) : C3", [ 96, 204 ] ],
      [ 128, [ 4, 32, 17, 2 ], "(D8 x D8) : C2", [ 128, 928 ] ],
      [ 129, [ 4, 32, 18, 2 ], "(((C2 x C2 x C2 x C2) : C2) : C2) : C3", [ 192, 201 ] ],
      [ 130, [ 4, 32, 18, 3 ], "(((C2 x C2 x C2 x C2) : C2) : C2) : C3", [ 192, 201 ] ],
      [ 131, [ 4, 32, 19, 2 ], "(((C2 x C2 x C2) : (C2 x C2)) : C3) : C2", [ 192, 1493 ] ],
      [ 132, [ 4, 32, 19, 3 ], "(((C2 x C2 x C2) : (C2 x C2)) : C3) : C2", [ 192, 1493 ] ],
      [ 133, [ 4, 32, 20, 2 ], "(((C2 x C2 x C2) : (C2 x C2)) : C3) : C2", [ 192, 1494 ] ],
      [ 134, [ 4, 32, 20, 3 ], "(((C2 x C2 x C2) : (C2 x C2)) : C3) : C2", [ 192, 1494 ] ],
      [ 135, [ 4, 32, 21, 2 ], "((((C2 x C2 x C2) : (C2 x C2)) : C3) : C2) : C2", [ 384, 5602 ] ],
      [ 136, [ 4, 32, 21, 3 ], "((((C2 x C2 x C2) : (C2 x C2)) : C3) : C2) : C2", [ 384, 5602 ] ],
      [ 137, [ 4, 33, 1, 1 ], "C3 x Q8", [ 24, 11 ] ],
      [ 138, [ 4, 33, 3, 1 ], "SL(2,3)", [ 24, 3 ] ],
      [ 139, [ 4, 33, 4, 1 ], "(C3 x Q8) : C2", [ 48, 17 ] ],
      [ 140, [ 4, 33, 5, 1 ], "((C4 x C2) : C2) : C3", [ 48, 33 ] ],
      [ 141, [ 4, 33, 6, 1 ], "GL(2,3)", [ 48, 29 ] ],
      [ 142, [ 4, 33, 7, 1 ], "C3 x SL(2,3)", [ 72, 25 ] ],
      [ 143, [ 4, 33, 8, 1 ], "((C2 x C2 x C2) : (C2 x C2)) : C3", [ 96, 201 ] ],
      [ 144, [ 4, 33, 9, 1 ], "GL(2,3) : C2", [ 96, 193 ] ],
      [ 145, [ 4, 33, 10, 1 ], "SL(2,3) : C4", [ 96, 67 ] ],
      [ 146, [ 4, 33, 11, 1 ], "(C3 x SL(2,3)) : C2", [ 144, 125 ] ],
      [ 147, [ 4, 33, 12, 1 ], "(GL(2,3) : C2) : C2", [ 192, 988 ] ],
      [ 148, [ 4, 33, 13, 1 ], "((C2 x C2 x C2) : (C2 x C2)) : (C3 x C3)", [ 288, 860 ] ],
      [ 149, [ 4, 33, 14, 1 ], "((((C2 x C2 x C2) : (C2 x C2)) : C3) : C2) : C3", [ 576, 8277 ] ],
      [ 150, [ 4, 33, 14, 2 ], "((((C2 x C2 x C2) : (C2 x C2)) : C3) : C2) : C3", [ 576, 8277 ] ],
      [ 151, [ 4, 33, 15, 1 ], "(((C2 x C2 x C2) : (C2 x C2)) : (C3 x C3)) : C2", [ 576, 8282 ] ],
      [ 152, [ 4, 33, 16, 1 ], "((((C2 x C2 x C2) : (C2 x C2)) : (C3 x C3)) : C2) : C2",
                               [ 1152, 157478 ] ] ]

######  Example 2.5 (The groups $I_{4,i}$ as in Table $9$).

    gap> Read("BCAlgTori.gap");
    gap> I4table:=List(I4,x->[Position(I4,x),x,
    > StructureDescription(MatGroupZClass(x[1],x[2],x[3],x[4])),
    > IdSmallGroup(MatGroupZClass(x[1],x[2],x[3],x[4]))]);
    [ [ 1, [ 4, 31, 1, 3 ], "C5 : C4", [ 20, 3 ] ],
      [ 2, [ 4, 31, 1, 4 ], "C5 : C4", [ 20, 3 ] ],
      [ 3, [ 4, 31, 2, 2 ], "C2 x (C5 : C4)", [ 40, 12 ] ],
      [ 4, [ 4, 31, 4, 2 ], "S5", [ 120, 34 ] ],
      [ 5, [ 4, 31, 5, 2 ], "S5", [ 120, 34 ] ],
      [ 6, [ 4, 31, 7, 2 ], "C2 x S5", [ 240, 189 ] ],
      [ 7, [ 4, 33, 2, 1 ], "C3 : C8", [ 24, 1 ] ] ]

##  3\. The subgroups $N_{3,j}\leq N_{3,i}$, $N_{31,j}\leq N_{31,i}$,
$N_{31,l},N_{4,j},I_{4,m}\leq N_{4,i}$ and $I_{4,j}\leq I_{4,i}$

We give all of the subgroups $N_{3,j}\leq N_{3,i}$ $(1\leq i,j\leq 15)$,
$N_{31,j}\leq N_{31,i}$ $(1\leq i,j\leq 64)$, $N_{31,l},N_{4,j},I_{4,m}\leq
N_{4,i}$ $(1\leq i,j\leq 152,1\leq l\leq 64,1\leq m\leq 7)$ and $I_{4,j}\leq
I_{4,i}$ $(1\leq i,j\leq 7)$ in Table $10$, Table $11$ and Table $12$
respectively where $N_{3,i}$, $N_{31,i}$, $N_{4,i}$ and $I_{4,i}$ are given as
in Definition 1.18.

Table $10$: The groups $N_{3,j}\leq N_{3,i}$ $(1\leq i,j\leq 15)$ (see Theorem
1.23 for the “bold” cases)

$i$ | $N_{3,j}\leq N_{3,i}$
---|---
$1$ | $\bm{N_{3,1}}$
$2$ | $\bm{N_{3,1}},\bm{N_{3,2}}$
$3$ | $\bm{N_{3,3}}$
$4$ | $\bm{N_{3,4}}$
$5$ | $\bm{N_{3,1}},\bm{N_{3,5}}$
$6$ | $\bm{N_{3,1}},\bm{N_{3,6}}$
$7$ | $N_{3,1},\bm{N_{3,2}},\bm{N_{3,3}},\bm{N_{3,4}},\bm{N_{3,5}},\bm{N_{3,6}},\bm{N_{3,7}}$
$8$ | $N_{3,1},\bm{N_{3,8}}$
$9$ | $N_{3,1},N_{3,2},N_{3,8},\bm{N_{3,9}}$
$10$ | $N_{3,3},\bm{N_{3,10}}$
$11$ | $N_{3,1},N_{3,5},N_{3,8},\bm{N_{3,11}}$
$12$ | $N_{3,1},N_{3,5},\bm{N_{3,12}}$
$13$ | $N_{3,1},N_{3,6},N_{3,8},\bm{N_{3,13}}$
$14$ | $N_{3,1},N_{3,2},N_{3,3},N_{3,4},N_{3,5},N_{3,6},N_{3,7},N_{3,8},N_{3,9},N_{3,11},N_{3,13},\bm{N_{3,14}}$
$15$ | $N_{3,1},N_{3,2},N_{3,3},N_{3,4},N_{3,5},N_{3,6},N_{3,7},\bm{N_{3,10}},\bm{N_{3,12}},\bm{N_{3,15}}$

Table $11$: The groups $N_{31,j}\leq N_{31,i}$ $(1\leq i,j\leq 64)$

$i$ | $N_{31,j}\leq N_{31,i}$
---|---
$1$ | $N_{31,1}$
$2$ | $N_{31,2}$
$3$ | $N_{31,1},N_{31,3}$
$4$ | $N_{31,4}$
$5$ | $N_{31,5}$
$6$ | $N_{31,4},N_{31,6}$
$7$ | $N_{31,7},N_{31,}$
$8$ | $N_{31,4},N_{31,8}$
$9$ | $N_{31,9}$
$10$ | $N_{31,1},N_{31,10}$
$11$ | $N_{31,1},N_{31,4},N_{31,11}$
$12$ | $N_{31,2},N_{31,5},N_{31,7},N_{31,9},N_{31,12}$
$13$ | $N_{31,1},N_{31,3},N_{31,4},N_{31,6},N_{31,8},N_{31,10},N_{31,11},N_{31,13}$
$14$ | $N_{31,14}$
$15$ | $N_{31,1},N_{31,15}$
$16$ | $N_{31,1},N_{31,16}$
$17$ | $N_{31,1},N_{31,2},N_{31,3},N_{31,14},N_{31,15},N_{31,16},N_{31,17}$
$18$ | $N_{31,18}$
$19$ | $N_{31,4},N_{31,19}$
$20$ | $N_{31,1},N_{31,20}$
$21$ | $N_{31,4},N_{31,21}$
$22$ | $N_{31,1},N_{31,22}$
$23$ | $N_{31,1},N_{31,3},N_{31,5},N_{31,18},N_{31,20},N_{31,22},N_{31,23}$
$24$ | $N_{31,2},N_{31,4},N_{31,6},N_{31,18},N_{31,19},N_{31,21},N_{31,24}$
$25$ | $N_{31,25}$
$26$ | $N_{31,26}$
$27$ | $N_{31,4},N_{31,27}$
$28$ | $N_{31,4},N_{31,28}$
$29$ | $N_{31,14},N_{31,18},N_{31,25},N_{31,26},N_{31,29}$
$30$ | $N_{31,4},N_{31,7},N_{31,8},N_{31,19},N_{31,25},N_{31,28},N_{31,30}$
$31$ | $N_{31,1},N_{31,4},N_{31,11},N_{31,15},N_{31,19},N_{31,20},N_{31,27},N_{31,31}$
$32$ | $N_{31,4},N_{31,8},N_{31,9},N_{31,21},N_{31,26},N_{31,27},N_{31,32}$
$33$ | $N_{31,1},N_{31,7},N_{31,10},N_{31,15},N_{31,22},N_{31,26},N_{31,33}$
$34$ | $N_{31,1},N_{31,9},N_{31,10},N_{31,16},N_{31,20},N_{31,25},N_{31,34}$
$35$ | $N_{31,1},N_{31,4},N_{31,11},N_{31,16},N_{31,21},N_{31,22},N_{31,28},N_{31,35}$
$36$ | $N_{31,4},N_{31,5},N_{31,6},N_{31,14},N_{31,27},N_{31,28},N_{31,36}$
$37$ |  $N_{31,1},N_{31,2},N_{31,3},N_{31,4},N_{31,5},N_{31,6},N_{31,7},N_{31,8},N_{31,9},N_{31,10},N_{31,11},N_{31,12},N_{31,13}$,
|
$N_{31,14},N_{31,15},N_{31,16},N_{31,17},N_{31,18},N_{31,19},N_{31,20},N_{31,21},N_{31,22},N_{31,23},N_{31,24},N_{31,25}$,
|
$N_{31,26},N_{31,27},N_{31,28},N_{31,29},N_{31,30},N_{31,31},N_{31,32},N_{31,33},N_{31,34},N_{31,35},N_{31,36},N_{31,37}$
$38$ | $N_{31,4},N_{31,38}$
$39$ | $N_{31,5},N_{31,39}$
$40$ | $N_{31,4},N_{31,6},N_{31,38},N_{31,40}$
$41$ | $N_{31,4},N_{31,19},N_{31,38},N_{31,41}$
$42$ | $N_{31,1},N_{31,22},N_{31,42}$
$43$ | $N_{31,4},N_{31,21},N_{31,38},N_{31,43}$
$44$ | $N_{31,1},N_{31,3},N_{31,5},N_{31,18},N_{31,20},N_{31,22},N_{31,23},N_{31,39},N_{31,42},N_{31,44}$
$45$ | $N_{31,2},N_{31,4},N_{31,6},N_{31,18},N_{31,19},N_{31,21},N_{31,24},N_{31,38},N_{31,40},N_{31,41},N_{31,43},N_{31,45}$
$46$ | $N_{31,7},N_{31,46}$
$47$ | $N_{31,4},N_{31,8},N_{31,38},N_{31,47}$
$48$ | $N_{31,1},N_{31,4},N_{31,11},N_{31,38},N_{31,48}$
$49$ | $N_{31,4},N_{31,28},N_{31,49}$
$50$ | $N_{31,4},N_{31,28},N_{31,38},N_{31,50}$
$51$ | $N_{31,4},N_{31,27},N_{31,38},N_{31,51}$
$52$ | $N_{31,2},N_{31,5},N_{31,7},N_{31,9},N_{31,12},N_{31,39},N_{31,46},N_{31,52}$
$53$ | $N_{31,1},N_{31,3},N_{31,4},N_{31,6},N_{31,8},N_{31,10},N_{31,11},N_{31,13},N_{31,38},N_{31,40},N_{31,47},N_{31,48},N_{31,53}$
$54$ | $N_{31,4},N_{31,5},N_{31,6},N_{31,14},N_{31,27},N_{31,28},N_{31,36},N_{31,39},N_{31,49},N_{31,54}$
$55$ | $N_{31,4},N_{31,5},N_{31,6},N_{31,14},N_{31,27},N_{31,28},N_{31,36},N_{31,38},N_{31,40},N_{31,50},N_{31,51},N_{31,55}$
$56$ | $N_{31,4},N_{31,7},N_{31,8},N_{31,19},N_{31,25},N_{31,28},N_{31,30},N_{31,46},N_{31,49},N_{31,56}$
$57$ | $N_{31,4},N_{31,7},N_{31,8},N_{31,19},N_{31,25},N_{31,28},N_{31,30},N_{31,38},N_{31,41},N_{31,47},N_{31,50},N_{31,57}$
$58$ | $N_{31,1},N_{31,7},N_{31,10},N_{31,15},N_{31,22},N_{31,26},N_{31,33},N_{31,42},N_{31,46},N_{31,58}$
$59$ | $N_{31,4},N_{31,8},N_{31,9},N_{31,21},N_{31,26},N_{31,27},N_{31,32},N_{31,38},N_{31,43},N_{31,47},N_{31,51},N_{31,59}$
$60$ | $N_{31,1},N_{31,4},N_{31,11},N_{31,15},N_{31,19},N_{31,20},N_{31,27},N_{31,31},N_{31,38},N_{31,41},N_{31,48},N_{31,51},N_{31,60}$
$61$ | $N_{31,1},N_{31,4},N_{31,11},N_{31,16},N_{31,21},N_{31,22},N_{31,28},N_{31,35},N_{31,42},N_{31,49},N_{31,61}$
$62$ | $N_{31,1},N_{31,4},N_{31,11},N_{31,16},N_{31,21},N_{31,22},N_{31,28},N_{31,35},N_{31,38},N_{31,43},N_{31,48},N_{31,50},N_{31,62}$
$63$ |  $N_{31,1},N_{31,2},N_{31,3},N_{31,4},N_{31,5},N_{31,6},N_{31,7},N_{31,8},N_{31,9},N_{31,10},N_{31,11},N_{31,12},N_{31,13},N_{31,14}$,
|
$N_{31,15},N_{31,16},N_{31,17},N_{31,18},N_{31,19},N_{31,20},N_{31,21},N_{31,22},N_{31,23},N_{31,24},N_{31,25},N_{31,26},N_{31,27}$,
|
$N_{31,28},N_{31,29},N_{31,30},N_{31,31},N_{31,32},N_{31,33},N_{31,34},N_{31,35},N_{31,36},N_{31,37},N_{31,39},N_{31,42},N_{31,44}$,
|
$N_{31,46},N_{31,49},N_{31,52},N_{31,54},N_{31,56},N_{31,58},N_{31,61},N_{31,63}$
$64$ |  $N_{31,1},N_{31,2},N_{31,3},N_{31,4},N_{31,5},N_{31,6},N_{31,7},N_{31,8},N_{31,9},N_{31,10},N_{31,11},N_{31,12},N_{31,13},N_{31,14}$,
|
$N_{31,15},N_{31,16},N_{31,17},N_{31,18},N_{31,19},N_{31,20},N_{31,21},N_{31,22},N_{31,23},N_{31,24},N_{31,25},N_{31,26},N_{31,27}$,
|
$N_{31,28},N_{31,29},N_{31,30},N_{31,31},N_{31,32},N_{31,33},N_{31,34},N_{31,35},N_{31,36},N_{31,37},N_{31,38},N_{31,40},N_{31,41}$,
|
$N_{31,43},N_{31,45},N_{31,47},N_{31,48},N_{31,50},N_{31,51},N_{31,53},N_{31,55},N_{31,57},N_{31,59},N_{31,60},N_{31,62},N_{31,64}$

Table $12$: The groups $N_{31,l},N_{4,j},I_{4,m}\leq N_{4,i}$ $(1\leq i,j\leq
152,1\leq l\leq 64,1\leq m\leq 7)$ (see Theorem 1.26 for the “bold” cases)

$i$ | $N_{31,j},N_{4,j},I_{4,j}\leq N_{4,i}$
---|---
$1$ | $\bm{N_{4,1}}$
$2$ | $\bm{N_{4,2}}$
$3$ | $\bm{N_{4,1}},\bm{N_{4,3}}$
$4$ | $\bm{N_{4,4}}$
$5$ | $\bm{N_{4,5}}$
$6$ | $\bm{N_{4,1}},\bm{N_{4,6}}$
$7$ | $\bm{N_{4,7}}$
$8$ | $\bm{N_{31,1}},\bm{N_{4,1}},\bm{N_{4,8}}$
$9$ | $\bm{N_{31,1}},\bm{N_{4,9}}$
$10$ | $N_{31,2},N_{4,2},N_{4,5},N_{4,7},\bm{N_{4,10}}$
$11$ | $N_{31,1},\bm{N_{31,2}},\bm{N_{31,3}},N_{4,1},\bm{N_{4,3}},\bm{N_{4,6}},\bm{N_{4,8}},\bm{N_{4,11}}$
$12$ | $N_{31,1},\bm{N_{31,3}},\bm{N_{4,4}},\bm{N_{4,9}},\bm{N_{4,12}}$
$13$ | $\bm{N_{4,13}}$
$14$ | $\bm{N_{4,14}}$
$15$ | $\bm{N_{4,1}},\bm{N_{4,15}}$
$16$ | $\bm{N_{31,1}},\bm{N_{4,16}}$
$17$ | $\bm{N_{4,1}},\bm{N_{4,17}}$
$18$ | $\bm{N_{31,1}},\bm{N_{4,18}}$
$19$ | $N_{31,1},\bm{N_{31,3}},\bm{N_{4,2}},\bm{N_{4,13}},\bm{N_{4,16}},\bm{N_{4,19}}$
$20$ | $\bm{N_{31,2}},N_{4,1},\bm{N_{4,3}},\bm{N_{4,13}},\bm{N_{4,15}},\bm{N_{4,17}},\bm{N_{4,20}}$
$21$ | $\bm{N_{31,1}},N_{31,3},\bm{N_{4,1}},N_{4,3},N_{4,14},N_{4,18},\bm{N_{4,21}}$
$22$ | $\bm{N_{31,2}},\bm{N_{4,4}},\bm{N_{4,14}},\bm{N_{4,22}}$
$23$ | $\bm{N_{4,23}}$
$24$ | $\bm{N_{4,24}}$
$25$ | $\bm{N_{4,1}},\bm{N_{4,25}}$
$26$ | $\bm{N_{4,1}},\bm{N_{4,26}}$
$27$ | $N_{31,2},N_{31,14},N_{4,13},N_{4,23},N_{4,24},\bm{N_{4,27}}$
$28$ | $\bm{N_{31,1}},\bm{N_{31,3}},\bm{N_{31,14}},\bm{N_{4,14}},\bm{N_{4,28}}$
$29$ | $N_{4,1},\bm{N_{4,5}},\bm{N_{4,6}},\bm{N_{4,15}},\bm{N_{4,23}},\bm{N_{4,26}},\bm{N_{4,29}}$
$30$ | $N_{4,1},\bm{N_{4,6}},\bm{N_{4,7}},\bm{N_{4,17}},\bm{N_{4,24}},\bm{N_{4,25}},\bm{N_{4,30}}$
$31$ | $N_{31,1},\bm{N_{31,15}},N_{4,1},\bm{N_{4,5}},\bm{N_{4,8}},\bm{N_{4,15}},\bm{N_{4,16}},\bm{N_{4,24}},\bm{N_{4,25}},\bm{N_{4,31}}$
$32$ | $\bm{N_{31,1}},\bm{N_{31,15}},\bm{N_{4,9}},\bm{N_{4,18}},\bm{N_{4,32}}$
$33$ | $N_{31,1},\bm{N_{31,16}},N_{4,1},\bm{N_{4,7}},\bm{N_{4,8}},\bm{N_{4,16}},\bm{N_{4,17}},\bm{N_{4,23}},\bm{N_{4,26}},\bm{N_{4,33}}$
$34$ | $\bm{N_{31,1}},\bm{N_{31,16}},\bm{N_{4,9}},\bm{N_{4,18}},\bm{N_{4,34}}$
$35$ | $\bm{N_{31,14}},N_{4,1},\bm{N_{4,2}},\bm{N_{4,3}},\bm{N_{4,25}},\bm{N_{4,26}},\bm{N_{4,35}}$
$36$ | $N_{31,14},\bm{N_{4,1}},N_{4,3},N_{4,4},\bm{N_{4,36}}$
$37$ |  $N_{31,1},N_{31,2},N_{31,3},N_{31,14},N_{31,15},N_{31,16},\bm{N_{31,17}},N_{4,1},N_{4,2},N_{4,3},N_{4,5},N_{4,6},N_{4,7},N_{4,8},\bm{N_{4,10}}$,
|
$\bm{N_{4,11}},N_{4,13},N_{4,15},N_{4,16},N_{4,17},\bm{N_{4,19}},\bm{N_{4,20}},N_{4,23},N_{4,24},N_{4,25},N_{4,26},\bm{N_{4,27}},\bm{N_{4,29}},\bm{N_{4,30}},\bm{N_{4,31}}$,
| $\bm{N_{4,33}},\bm{N_{4,35}},\bm{N_{4,37}}$
$38$ |  $N_{31,1},\bm{N_{31,2}},\bm{N_{31,3}},\bm{N_{31,14}},\bm{N_{31,15}},\bm{N_{31,16}},N_{31,17},N_{4,1},\bm{N_{4,3}},\bm{N_{4,4}},\bm{N_{4,6}},\bm{N_{4,8}},\bm{N_{4,9}},N_{4,11},N_{4,12}$,
|
$\bm{N_{4,14}},\bm{N_{4,18}},N_{4,21},N_{4,22},N_{4,28},N_{4,32},N_{4,34},N_{4,36},\bm{N_{4,38}}$
$39$ | $\bm{N_{4,39}}$
$40$ | $\bm{N_{31,1}},\bm{N_{31,3}},\bm{N_{4,13}},\bm{N_{4,39}},\bm{N_{4,40}}$
$41$ | $N_{31,2},N_{4,14},N_{4,39},\bm{N_{4,41}}$
$42$ | $N_{4,4},N_{4,13},\bm{N_{4,42}}$
$43$ | $N_{4,1},\bm{N_{4,3}},\bm{N_{4,13}},\bm{N_{4,14}},\bm{N_{4,43}}$
$44$ | $N_{4,2},N_{4,14},\bm{N_{4,44}}$
$45$ | $\bm{N_{4,1}},N_{4,3},N_{4,39},\bm{N_{4,45}}$
$46$ | $\bm{N_{4,2}},\bm{N_{4,4}},\bm{N_{4,39}},\bm{N_{4,46}}$
$47$ | $N_{31,1},\bm{N_{31,3}},\bm{N_{4,2}},\bm{N_{4,4}},\bm{N_{4,9}},N_{4,12},\bm{N_{4,13}},\bm{N_{4,16}},N_{4,19},\bm{N_{4,39}},N_{4,40},N_{4,42},N_{4,46},\bm{N_{4,47}}$
$48$ |  $N_{31,1},\bm{N_{31,2}},\bm{N_{31,3}},N_{4,1},\bm{N_{4,3}},\bm{N_{4,6}},\bm{N_{4,8}},N_{4,11},\bm{N_{4,13}},\bm{N_{4,14}},\bm{N_{4,15}},\bm{N_{4,17}},\bm{N_{4,18}},N_{4,20},N_{4,21}$,
| $\bm{N_{4,39}},N_{4,40},N_{4,41},N_{4,43},N_{4,45},\bm{N_{4,48}}$
$49$ | $N_{31,2},N_{4,2},N_{4,4},N_{4,5},N_{4,7},N_{4,10},N_{4,14},N_{4,22},N_{4,39},N_{4,41},N_{4,44},N_{4,46},\bm{N_{4,49}}$
$50$ | $\bm{N_{31,14}},\bm{N_{4,13}},\bm{N_{4,14}},\bm{N_{4,50}}$
$51$ | $\bm{N_{31,14}},\bm{N_{4,39}},\bm{N_{4,51}}$
$52$ |  $N_{31,1},\bm{N_{31,2}},\bm{N_{31,3}},\bm{N_{31,14}},\bm{N_{31,15}},\bm{N_{31,16}},N_{31,17},\bm{N_{4,13}},\bm{N_{4,14}},\bm{N_{4,23}},\bm{N_{4,24}},N_{4,27},N_{4,28},\bm{N_{4,39}}$,
| $N_{4,40},N_{4,41},N_{4,50},N_{4,51},\bm{N_{4,52}}$
$53$ |  $N_{31,1},\bm{N_{31,3}},\bm{N_{31,14}},N_{4,1},\bm{N_{4,2}},\bm{N_{4,3}},\bm{N_{4,13}},\bm{N_{4,14}},\bm{N_{4,16}},\bm{N_{4,18}},N_{4,19},N_{4,21},\bm{N_{4,25}},\bm{N_{4,26}},N_{4,28}$,
| $N_{4,35},N_{4,43},N_{4,44},N_{4,50},\bm{N_{4,53}}$
$54$ |  $N_{31,2},N_{31,14},N_{4,1},N_{4,3},N_{4,4},N_{4,13},N_{4,14},N_{4,15},N_{4,17},\bm{N_{4,20}},\bm{N_{4,22}},N_{4,23},N_{4,24},\bm{N_{4,27}},\bm{N_{4,36}}$,
| $\bm{N_{4,42}},\bm{N_{4,43}},\bm{N_{4,50}},\bm{N_{4,54}}$
$55$ | $N_{31,14},N_{4,1},N_{4,2},N_{4,3},N_{4,4},N_{4,25},N_{4,26},\bm{N_{4,35}},\bm{N_{4,36}},N_{4,39},\bm{N_{4,45}},\bm{N_{4,46}},\bm{N_{4,51}},\bm{N_{4,55}}$
$56$ |  $N_{31,1},N_{31,2},N_{31,3},N_{31,14},N_{31,15},N_{31,16},\bm{N_{31,17}},N_{4,1},N_{4,2},N_{4,3},N_{4,4},N_{4,5},N_{4,6},N_{4,7},N_{4,8},N_{4,9}$,
|
$\bm{N_{4,10}},\bm{N_{4,11}},\bm{N_{4,12}},N_{4,13},N_{4,14},N_{4,15},N_{4,16},N_{4,17},N_{4,18},\bm{N_{4,19}},\bm{N_{4,20}},\bm{N_{4,21}},\bm{N_{4,22}},N_{4,23},N_{4,24}$,
|
$N_{4,25},N_{4,26},\bm{N_{4,27}},\bm{N_{4,28}},\bm{N_{4,29}},\bm{N_{4,30}},\bm{N_{4,31}},\bm{N_{4,32}},\bm{N_{4,33}},\bm{N_{4,34}},\bm{N_{4,35}},\bm{N_{4,36}},N_{4,37},N_{4,38},N_{4,39}$,
|
$\bm{N_{4,40}},\bm{N_{4,41}},\bm{N_{4,42}},\bm{N_{4,43}},\bm{N_{4,44}},\bm{N_{4,45}},\bm{N_{4,46}},N_{4,47},N_{4,48},N_{4,49},\bm{N_{4,50}},\bm{N_{4,51}},N_{4,52},N_{4,53},N_{4,54}$,
| $N_{4,55},\bm{N_{4,56}}$
$57$ | $\bm{N_{4,57}}$
$58$ | $N_{4,57},\bm{N_{4,58}}$
$59$ | $N_{4,57},\bm{N_{4,59}}$
$60$ | $N_{4,57},\bm{N_{4,60}}$
$61$ | $N_{4,57},\bm{N_{4,61}}$
$62$ | $N_{4,57},\bm{N_{4,62}}$
$63$ | $N_{4,57},\bm{N_{4,58}},\bm{N_{4,59}},\bm{N_{4,60}},\bm{N_{4,63}}$
$64$ | $N_{4,57},\bm{N_{4,58}},\bm{N_{4,61}},\bm{N_{4,62}},\bm{N_{4,64}}$
$65$ | $N_{4,57},N_{4,59},N_{4,51},\bm{N_{4,65}}$
$66$ | $N_{4,57},\bm{N_{4,59}},\bm{N_{4,60}},\bm{N_{4,62}},\bm{N_{4,66}}$
$67$ | $N_{4,57},N_{4,60},N_{4,61},\bm{N_{4,67}}$
$68$ | $N_{4,57},N_{4,57},N_{4,58},N_{4,59},N_{4,60},N_{4,61},N_{4,62},\bm{N_{4,63}},\bm{N_{4,64}},\bm{N_{4,65}},\bm{N_{4,66}},\bm{N_{4,67}},\bm{N_{4,68}}$
$69$ | $N_{4,2},\bm{N_{4,69}}$
$70$ | $N_{4,4},\bm{N_{4,70}}$
$71$ | $N_{31,1},N_{4,16},\bm{N_{4,71}}$
$72$ | $N_{31,1},N_{31,3},N_{4,2},N_{4,13},N_{4,16},N_{4,19},\bm{N_{4,69}},\bm{N_{4,71}},\bm{N_{4,72}}$
$73$ | $N_{31,2},N_{4,4},N_{4,14},N_{4,22},N_{4,70},\bm{N_{4,73}}$
$74$ | $N_{4,5},\bm{N_{4,74}}$
$75$ | $N_{4,7},\bm{N_{4,75}}$
$76$ | $N_{31,1},N_{4,9},\bm{N_{4,76}}$
$77$ | $N_{4,1},N_{4,26},\bm{N_{4,77}}$
$78$ | $N_{4,1},N_{4,25},\bm{N_{4,78}}$
$79$ | $N_{31,2},N_{4,2},N_{4,5},N_{4,7},N_{4,10},N_{4,69},N_{4,74},N_{4,75},\bm{N_{4,79}}$
$80$ | $N_{31,1},N_{31,3},N_{4,4},N_{4,9},N_{4,12},\bm{N_{4,70}},\bm{N_{4,76}},\bm{N_{4,80}}$
$81$ | $N_{31,14},N_{4,1},N_{4,2},N_{4,3},N_{4,25},N_{4,26},N_{4,35},N_{4,69},N_{4,77},N_{4,78},\bm{N_{4,81}}$
$82$ | $N_{31,14},N_{4,1},N_{4,3},N_{4,4},N_{4,36},N_{4,70},\bm{N_{4,82}}$
$83$ | $N_{4,1},N_{4,5},N_{4,6},N_{4,15},N_{4,23},N_{4,26},N_{4,29},\bm{N_{4,74}},\bm{N_{4,77}},\bm{N_{4,83}}$
$84$ | $N_{31,1},N_{31,15},N_{4,1},N_{4,5},N_{4,8},N_{4,15},N_{4,16},N_{4,24},N_{4,25},N_{4,31},N_{4,71},N_{4,74},N_{4,78},\bm{N_{4,84}}$
$85$ | $N_{4,1},N_{4,6},N_{4,7},N_{4,17},N_{4,24},N_{4,25},N_{4,30},\bm{N_{4,75}},\bm{N_{4,78}},\bm{N_{4,85}}$
$86$ | $N_{31,1},\bm{N_{31,15}},\bm{N_{4,9}},\bm{N_{4,18}},N_{4,32},N_{4,76},\bm{N_{4,86}}$
$87$ | $N_{31,1},N_{31,16},N_{4,1},N_{4,7},N_{4,8},N_{4,16},N_{4,17},N_{4,23},N_{4,26},N_{4,33},N_{4,71},N_{4,75},N_{4,77},\bm{N_{4,87}}$
$88$ | $N_{31,1},\bm{N_{31,16}},\bm{N_{4,9}},\bm{N_{4,18}},N_{4,34},N_{4,76},\bm{N_{4,88}}$
$89$ |  $N_{31,1},N_{31,2},N_{31,3},N_{31,14},N_{31,15},N_{31,16},N_{31,17},N_{4,1},N_{4,2},N_{4,3},N_{4,5},N_{4,6},N_{4,7},N_{4,8},N_{4,10}$,
|
$N_{4,11},N_{4,13},N_{4,15},N_{4,16},N_{4,17},N_{4,19},N_{4,20},N_{4,23},N_{4,24},N_{4,25},N_{4,26},N_{4,27},N_{4,29},N_{4,30},N_{4,31}$,
|
$N_{4,33},N_{4,35},N_{4,37},N_{4,69},N_{4,71},\bm{N_{4,72}},N_{4,74},N_{4,75},N_{4,77},N_{4,78},\bm{N_{4,79}},\bm{N_{4,81}},\bm{N_{4,83}},\bm{N_{4,84}},\bm{N_{4,85}}$,
| $\bm{N_{4,87}},\bm{N_{4,89}}$
$90$ |  $N_{31,1},N_{31,2},N_{31,3},N_{31,14},N_{31,15},N_{31,16},N_{31,17},N_{4,1},N_{4,3},N_{4,4},N_{4,6},N_{4,8},N_{4,9},N_{4,11},N_{4,12}$,
|
$N_{4,14},N_{4,18},N_{4,21},N_{4,22},N_{4,28},N_{4,32},N_{4,34},N_{4,36},N_{4,38},N_{4,70},\bm{N_{4,73}},N_{4,76},\bm{N_{4,80}},\bm{N_{4,82}},\bm{N_{4,86}}$,
| $\bm{N_{4,88}},\bm{N_{4,90}}$
$91$ | $N_{4,57},\bm{N_{4,91}}$
$92$ | $N_{4,57},\bm{N_{4,92}}$
$93$ | $N_{4,57},\bm{N_{4,58}},\bm{N_{4,91}},\bm{N_{4,92}},\bm{N_{4,93}}$
$94$ | $N_{4,57},N_{4,61},N_{4,91},\bm{N_{4,94}}$
$95$ | $N_{4,57},N_{4,61},N_{4,92},\bm{N_{4,95}}$
$96$ | $N_{4,57},\bm{N_{4,62}},\bm{N_{4,91}},\bm{N_{4,92}},\bm{N_{4,96}}$
$97$ | $N_{4,57},N_{4,61},\bm{N_{4,97}}$
$98$ | $N_{4,57},N_{4,61},\bm{N_{4,98}}$
$99$ | $N_{4,57},N_{4,58},N_{4,61},N_{4,62},\bm{N_{4,64}},N_{4,91},N_{4,92},\bm{N_{4,93}},\bm{N_{4,94}},\bm{N_{4,95}},\bm{N_{4,96}},\bm{N_{4,99}}$
$100$ | $N_{4,57},N_{4,58},N_{4,61},N_{4,62},\bm{N_{4,64}},\bm{N_{4,97}},\bm{N_{4,98}},\bm{N_{4,100}}$
$101$ | $N_{4,57},N_{4,59},N_{4,61},N_{4,65},N_{4,91},N_{4,94},N_{4,97},\bm{N_{4,101}}$
$102$ | $N_{4,57},N_{4,59},N_{4,61},\bm{N_{4,65}},N_{4,92},\bm{N_{4,95}},\bm{N_{4,98}},\bm{N_{4,102}}$
$103$ | $N_{4,57},N_{4,60},N_{4,61},\bm{N_{4,67}},N_{4,91},\bm{N_{4,94}},\bm{N_{4,98}},\bm{N_{4,103}}$
$104$ | $N_{4,57},N_{4,60},N_{4,61},N_{4,67},N_{4,92},N_{4,95},N_{4,97},\bm{N_{4,104}}$
$105$ |  $N_{4,57},N_{4,58},N_{4,59},N_{4,60},N_{4,61},N_{4,62},N_{4,63},N_{4,64},N_{4,65},N_{4,66},N_{4,67},\bm{N_{4,68}},N_{4,91},N_{4,92},N_{4,93}$,
|
$N_{4,94},N_{4,95},N_{4,96},N_{4,97},N_{4,98},\bm{N_{4,99}},\bm{N_{4,100}},\bm{N_{4,101}},\bm{N_{4,102}},\bm{N_{4,103}},\bm{N_{4,104}},\bm{N_{4,105}}$
$106$ | $\bm{N_{4,106}}$
$107$ | $N_{4,39},\bm{N_{4,107}}$
$108$ | $N_{4,106},\bm{N_{4,108}}$
$109$ | $N_{4,39},N_{4,106},\bm{N_{4,109}}$
$110$ | $N_{4,106},\bm{N_{4,110}}$
$111$ | $N_{4,106},\bm{N_{4,111}}$
$112$ | $\bm{N_{4,1}},N_{4,3},N_{4,39},N_{4,45},N_{4,106},N_{4,107},N_{4,108},N_{4,109},\bm{N_{4,112}}$
$113$ | $\bm{N_{4,2}},\bm{N_{4,4}},\bm{N_{4,39}},N_{4,46},N_{4,107},\bm{N_{4,113}}$
$114$ | $N_{31,14},N_{4,39},N_{4,51},N_{4,106},N_{4,107},N_{4,109},\bm{N_{4,114}}$
$115$ | $N_{4,2},N_{4,4},N_{4,13},N_{4,39},N_{4,42},N_{4,46},\bm{N_{4,115}}$
$116$ | $N_{4,2},N_{4,4},N_{4,14},N_{4,39},N_{4,44},N_{4,46},\bm{N_{4,116}}$
$117$ | $\bm{N_{4,2}},\bm{N_{4,4}},\bm{N_{4,39}},N_{4,46},\bm{N_{4,106}},N_{4,109},\bm{N_{4,117}}$
$118$ | $N_{4,106},N_{4,108},N_{4,110},\bm{N_{4,118}}$
$119$ | $N_{4,106},N_{4,108},N_{4,111},\bm{N_{4,119}}$
$120$ |  $N_{31,14},N_{4,1},N_{4,2},N_{4,3},N_{4,4},N_{4,25},N_{4,26},\bm{N_{4,35}},\bm{N_{4,36}},N_{4,39},\bm{N_{4,45}},\bm{N_{4,46}},\bm{N_{4,51}},N_{4,55},N_{4,106}$,
|
$\bm{N_{4,107}},\bm{N_{4,108}},\bm{N_{4,109}},N_{4,112},N_{4,113},N_{4,114},N_{4,117},\bm{N_{4,120}}$
$121$ |  $N_{31,2},N_{4,2},N_{4,4},N_{4,5},N_{4,7},N_{4,10},N_{4,13},N_{4,14},N_{4,22},N_{4,39},N_{4,41},N_{4,42},N_{4,44},N_{4,46},N_{4,49}$,
| $N_{4,107},N_{4,113},N_{4,115},\bm{N_{4,121}}$
$122$ |  $N_{31,1},\bm{N_{31,3}},\bm{N_{4,2}},\bm{N_{4,4}},\bm{N_{4,9}},N_{4,12},\bm{N_{4,13}},\bm{N_{4,14}},\bm{N_{4,16}},N_{4,19},\bm{N_{4,39}},N_{4,40},N_{4,42},N_{4,44},N_{4,46}$,
| $N_{4,47},N_{4,107},N_{4,113},N_{4,116},\bm{N_{4,122}}$
$123$ |  $N_{31,1},\bm{N_{31,3}},\bm{N_{4,2}},\bm{N_{4,4}},\bm{N_{4,9}},N_{4,12},\bm{N_{4,13}},\bm{N_{4,16}},N_{4,19},\bm{N_{4,39}},N_{4,40},N_{4,42},N_{4,46},N_{4,47},\bm{N_{4,106}}$,
| $N_{4,109},N_{4,115},N_{4,117},\bm{N_{4,123}}$