Table 6. Data for sporadic $3$-groups $G$ with $11\leq n=\mathrm{lo}(G)\leq 12$ of the forest $\mathcal{F}(5)$ $\\#$ | $m,n$ | $\rho;\alpha,\beta,\gamma,\delta$ | dp | dl | $\zeta$ | $\mu$ | $\nu$ | $\tau(1)$ | $\tau_{2}$ | Type | $\varkappa$ | $\sigma$ | $\\#\mathrm{Aut}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$1$ | $6,9$ |  $0;0,0,0,0$ ($P_{9}$) | 0 | 2 | $1^{2}$ | $6$ | $2$ | $32$ | $2^{3}1$ | $\mathrm{b}.10^{\ast}$ | $(0043)$ | $2$ | $2^{3}\cdot 3^{14}$
$1$ | $7,11$ |  $0;0,0,0,0$ ($P_{11}$) | 0 | 2 | $1^{2}$ | $6$ | $2$ | $3^{2}$ | $32^{3}$ | $\mathrm{b}.10^{\ast}$ | $(0043)$ | $2$ | $2^{3}\cdot 3^{18}$
$1$ | $7,11$ | $0;0,1,0,1$ | 0 | 2 | $1^{2}$ | $5$ | $1$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.19^{\ast}$ | $(0343)$ | $0$ | $3^{18}$
$1$ | $7,11$ | $0;0,0,0,1$ | 0 | 2 | $1^{2}$ | $5$ | $1$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.23^{\ast}$ | $(0243)$ | $0$ | $2\cdot 3^{18}$
$1$ | $7,11$ | $0;0,1,0,0$ | 0 | 2 | $1^{2}$ | $5$ | $1$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.25^{\ast}$ | $(0143)$ | $0$ | $2\cdot 3^{18}$
$1$ | $7,11$ | $0;1,1,-1,1$ | 0 | 2 | $1^{2}$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{F}.7$ | $(3443)$ | $0$ | $3^{18}$
$2$ | $7,11$ | $0;1,\pm 1,0,0$ | 0 | 2 | $1^{2}$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{F}.11$ | $(1143)$ | $0$ | $2\cdot 3^{18}$
$2$ | $7,11$ | $0;\pm(1,0,1),1$ | 0 | 2 | $1^{2}$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{F}.12$ | $(1343)$ | $0$ | $3^{18}$
$2$ | $7,11$ | $0;1,\pm(1,1),0$ | 0 | 2 | $1^{2}$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{F}.13$ | $(3143)$ | $0$ | $3^{18}$
$1$ | $7,11$ | $0;1,0,0,1$ | 0 | 2 | $1^{2}$ | $5$ | $1$ | $3^{2}$ | $32^{3}$ | $\mathrm{G}.16$ | $(1243)$ | $0$ | $2\cdot 3^{18}$
$1$ | $7,11$ | $0;0,1,1,0$ | 0 | 2 | $1^{2}$ | $5$ | $1$ | $3^{2}$ | $32^{3}$ | $\mathrm{G}.19$ | $(2143)$ | $0$ | $2\cdot 3^{18}$
$2$ | $7,11$ | $0;1,\pm(1,1),1$ | 0 | 2 | $1^{2}$ | $5$ | $1$ | $3^{2}$ | $32^{3}$ | $\mathrm{H}.4$ | $(3343)$ | $0$ | $2\cdot 3^{18}$
$12$ | $8,12$ |  | 1 | 2 | $1$ | $4$ | $0$ | $3^{2}$ | $3^{2}2^{2}$ |  |  | $0$ | $2\cdot 3^{20}$
$12$ | $8,12$ |  | 1 | 2 | $1$ | $4$ | $0$ | $3^{2}$ | $3^{2}2^{2}$ |  |  | $0$ | $3^{20}$
$8$ | $8,12$ |  | 1 | 3 | $1$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ |  |  | $0$ | $2\cdot 3^{19}$
$20$ | $8,12$ |  | 1 | 3 | $1$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ |  |  | $0$ | $3^{19}$
$8$ | $8,12$ |  | 1 | 3 | $1$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ |  |  | $0$ | $3^{18}$

##  12\. Sporadic and periodic $3$-groups $G$ of odd coclass
$\mathrm{cc}(G)\geq 5$

Although formulated for the particular coclass $r=5$, all results on sporadic
and periodic groups in this section are valid for any odd coclass $r\geq 5$.
The exemplary (co-periodic) sporadic part $\mathcal{F}_{0}(5)$ of the coclass
forest $\mathcal{F}(5)$ is presented in the following Proposition 12.1.

######  Proposition 12.1.

The sporadic part $\mathcal{F}_{0}(5)$ of the coclass-$5$ forest
$\mathcal{F}(5)$ consists of

  * •

$7$ $(1+2+2+2)$ isolated metabelian vertices with types $\mathrm{F}.7$,
$\mathrm{F}.11$, $\mathrm{F}.12$, $\mathrm{F}.13$,

  * •

$4$ $(1+1+2)$ metabelian roots of finite trees with types $\mathrm{G}.16$,
$\mathrm{G}.19$, $\mathrm{H}.4$, together with their $24$ metabelian and $36$
non-metabelian children, all with depth $\mathrm{dp}=1$,

  * •

$34$ $(16+9+9)$ isolated vertices with $\mathrm{dl}=3$ and types
$\mathrm{d}.19$, $\mathrm{d}.23$, $\mathrm{d}.25$,

  * •

$89$ isolated vertices with $\mathrm{dl}=3$ and type $\mathrm{b}.10$,

  * •

$13$ capable vertices with $\mathrm{dl}=3$ and type $\mathrm{b}.10$,
whose children do not belong to $\mathcal{F}_{0}(5)$, by definition.

The action flag of all vertices is $\sigma=0$, and consequently none of them
has an RI- or $V_{4}$-action.

Together with the $4$ metabelian roots of coclass-$5$ trees, the
$7+4+34+89+13=11+136$ vertices of depth $\mathrm{dp}=0$ are exactly the
$N_{2}=151$ children of step size $s=2$ of $P_{9}=\langle
2187,64\rangle-\\#2;33$, and the $4+(4+13)$ capable vertices among them
correspond to the invariant $C_{2}=21$ of $P_{9}$.

Figure 12. Metabelian interface between the coclass forests $\mathcal{F}(4)$
and $\mathcal{F}(5)$order
$3^{n}$$2\,187$$3^{7}$$6\,561$$3^{8}$$19\,683$$3^{9}$$59\,049$$3^{10}$$177\,147$$3^{11}$$531\,441$$3^{12}$$1\,594\,323$$3^{13}$$4\,782\,969$$3^{14}$Legend:$\zeta=1^{2},\sigma=2$$\zeta=1^{2},\sigma=0$$\zeta=1,\sigma=0$Edge
of depth $2$ forming the interfacebetween $\mathcal{F}(3)$ and
$\mathcal{F}(4)$$\mathcal{T}^{4}P_{9}\subset\mathcal{F}(4)$Edges of depth $2$
forming the interfacebetween $\mathcal{F}(4)$ and
$\mathcal{F}(5)$$4$metabelian
mainlines$\mathcal{T}^{5}R_{2}^{5}$$\mathcal{T}^{5}R_{3}^{5}$$\mathcal{T}^{5}R_{4}^{5}$$\mathcal{T}^{5}R_{1}^{5}$$\langle
64\rangle=P_{7}\in\mathcal{F}(3)$$P_{9}=P_{7}-\\#2;33\in\mathcal{F}(4)$$\\#2;38$$\ast
2$$27$$34$$\ast 2$$33$$36$$\ast
2$$31$$35$$30$$6\ast$$26$$6\ast$$32$$6\ast$$28$$6\ast$$29=$$R_{2}^{5}$$37=$$R_{3}^{5}$$39=$$R_{4}^{5}$$25=$$R_{1}^{5}=P_{11}$TKTF.7F.11F.12F.13$\varkappa=$$(3443)$$(1143)$$(1343)$$(3143)$TKTG.16G.19H.4$\varkappa=$$(1243)$$(2143)$$(3343)$TKTd.19∗d.23∗d.25∗b.10∗$\varkappa=$$(0343)$$(0243)$$(0143)$$(0043)$

Figure 12 sketches an outline of the metabelian skeleton of the coclass forest
$\mathcal{F}(5)$ in its top region. The vertices $P_{7}=\langle
2187,64\rangle\in\mathcal{F}(3)$ and $P_{9}=\langle
2187,64\rangle-\\#2;33\in\mathcal{F}(4)$, with the crucial bifurcation from
$\mathcal{F}(4)$ to $\mathcal{F}(5)$, belong to the infinite main trunk (§
10).

######  Theorem 12.1.

The coclass-$r$ forest $\mathcal{F}(r)$ with any odd $r\geq 5$ is the disjoint
union of its finite sporadic part $\mathcal{F}_{0}(r)$ with total information
content

(12.1) $s=\\#\mathcal{F}_{0}(r)=207$

and $t=4$ infinite coclass-$r$ trees $\mathcal{T}^{r}(R_{i}^{r})$ with roots
$R_{i}^{r}:=P_{2r-1}-\\#2;n_{i}$, where $(n_{i})_{1\leq i\leq
4}=(25,29,37,39)$ for $r=5$. The algebraic invariants for groups with centre
$\zeta=1^{2}$, and in cumulative form for $\zeta=1$, are given for $r=5$ in
Table 6, where the parent vertex $P_{2r-1}=P_{9}$ on the maintrunk is also
included, but the $136$ non-metabelian top vertices of depth $\mathrm{dp}=0$
are excluded.

###### Proof.

(of Proposition 12.1 and Theorem 12.1) We have computed the sporadic parts
$\mathcal{F}_{0}(r)$ of coclass forests $\mathcal{F}(r)$ with odd $r\geq 5$ up
to $r\leq 21$ by means of MAGMA [17]. They all share a common graph theoretic
structure with $\mathcal{F}_{0}(5)$. The forest $\mathcal{F}(5)$ contains $4$
roots of coclass trees with metabelian mainlines (a unique root $R_{1}^{5}$ of
type $\mathrm{b}$ and three roots $R_{2}^{5},\ldots,R_{4}^{5}$ of type
$\mathrm{d}$), namely
$R_{1}^{5}=P_{11}=G^{7,11}_{0}(0,0,0,0)\simeq P_{9}-\\#2;25$,
$R_{2}^{5}=G^{7,11}_{0}(0,1,0,1)\simeq P_{7}-\\#2;29$,
$R_{3}^{5}=G^{7,11}_{0}(0,0,0,1)\simeq P_{9}-\\#2;37$,
$R_{4}^{5}=G^{7,11}_{0}(0,1,0,0)\simeq P_{9}-\\#2;39$,
which give rise to the periodic part of $\mathcal{F}(5)$, and $35$ sporadic
metabelian groups of type $\mathrm{F}$, $\mathrm{G}$ or $\mathrm{H}$. Among
the groups of the sporadic part $\mathcal{F}_{0}(5)$, there are $7$ isolated
metabelian vertices with type $\mathrm{F}$, and $4$ metabelian roots of finite
trees with type $\mathrm{G}$ or $\mathrm{H}$ and tree depth $1$. Among the
$60=15+15+15+15$ children, there are $24=6+6+6+6$ metabelian, and $36=9+9+9+9$
have derived length $3$. The latter are omitted in the forest diagram, Figure
12. Additionally, $\mathcal{F}_{0}(5)$ contains $136$ non-metabelian top
vertices, which gives a total information content $s=\\#\mathcal{F}_{0}(5)$ of
$207=(11+136)+60$ representatives.

The metabelian skeleton consists of $35=11+24$ vertices. The results for
metabelian groups are in accordance with the third tree diagram $e\geq 4$,
$e\equiv 0\pmod{2}$, in [7, third page between pp. 191–192]. The metabelian
groups in Table 6 correspond to the representatives of isomorphism classes in
[28, pp. 34–35]. ∎

###  12.1. The unique mainline of type $\mathrm{b}.10^{\ast}$ for odd coclass
$r\geq 5$

######  Proposition 12.2.

(Periodicity and descendant numbers.)
The branches $\mathcal{B}(i)$, $i\geq n_{\ast}=11$, of the coclass-$5$ tree
$\mathcal{T}^{5}(\langle 2187,64\rangle-\\#2;33-\\#2;25)$ with mainline
vertices of transfer kernel type $\mathrm{b}.10^{\ast}$,
$\varkappa\sim(0043)$, are periodic with pre-period length $\ell_{\ast}=1$ and
with primitive period length $\ell=2$, that is,
$\mathcal{B}(i+2)\simeq\mathcal{B}(i)$ are isomorphic as digraphs, for all
$i\geq p_{\ast}=n_{\ast}+\ell_{\ast}=12$.

The graph theoretic structure of the tree is determined uniquely by the
numbers $N_{1}$ of immediate descendants and $C_{1}$ of capable immediate
descendants of the mainline vertices $m_{n}$ with logarithmic order
$n=\mathrm{lo}(m_{n})\geq n_{\ast}=11$:
$(N_{1},C_{1})=(30,1)$ for the root $m_{11}$ with $n=11$,
$(N_{1},C_{1})=(24,1)$ for all mainline vertices $m_{n}$ with even logarithmic
order $n\geq 12$,
$(N_{1},C_{1})=(40,1)$ for all mainline vertices $m_{n}$ with odd logarithmic
order $n\geq 13$.

###### Proof.

(of Proposition 12.2) The statements concerning the numbers $N_{1}(m_{n})$ of
immediate descendants of the mainline vertices $m_{n}$ with $n\geq
n_{\ast}=11$ have been obtained by direct computation with the $p$-group
generation algorithm [12, 13, 14] in MAGMA [17]. In detail, we proved that
there are
$6$, resp. $6$, metabelian vertices with bicyclic centre $\zeta=1^{2}$, resp.
cyclic centre $\zeta=1$, and
$9$, resp. $9$, non-metabelian vertices with $\zeta=1^{2}$, resp. $\zeta=1$,
together $30$ vertices ($12$ of them metabelian) in the pre-periodic branch
$\mathcal{B}(11)$,
and the primitive period $(\mathcal{B}(12),\mathcal{B}(13))$ of length
$\ell=2$ consists of
$4$, resp. $6$, metabelian vertices with $\zeta=1^{2}$, resp. $\zeta=1$, and
$5$, resp. $9$, non-metabelian vertices with $\zeta=1^{2}$, resp. $\zeta=1$,
together $24$ vertices ($10$ of them metabelian) in branch $\mathcal{B}(12)$,
and
$6$, resp. $9$, metabelian vertices with $\zeta=1^{2}$, resp. $\zeta=1$, and
$9$, resp. $16$, non-metabelian vertices with $\zeta=1^{2}$, resp. $\zeta=1$,
together $40$ vertices ($15$ of them metabelian) in branch $\mathcal{B}(13)$.

The results concerning the metabelian skeleton confirm the corresponding
statements in the dissertation of Nebelung [7, Thm. 5.1.16, pp. 178–179, and
the third Figure, $e\geq 4$, $e\equiv 0\pmod{2}$, on the third page between
pp. 191–192]. The tree $\mathcal{T}^{5}{P_{11}}$ corresponds to the infinite
metabelian pro-$3$ group $S_{4,1}$ in [6, Cnj. 15 (a), p. 116].

The claim of the virtual periodicity of branches has been proved generally for
any coclass tree in [1] and [2]. Here, the strict periodicity was confirmed by
computation up to branch $\mathcal{B}(33)$ and clearly sets in at
$p_{\ast}=12$. ∎

Figure 13. The unique coclass-$5$ tree $\mathcal{T}^{5}{P_{11}}$ with mainline
of type $\mathrm{b}.10^{\ast}$order
$3^{n}$$177\,147$$3^{11}$$531\,441$$3^{12}$$1\,594\,323$$3^{13}$$4\,782\,969$$3^{14}$$14\,348\,907$$3^{15}$$43\,046\,721$$3^{16}$$129\,140\,163$$3^{17}$$\tau(1)=$$(3^{2})$$(43)$$(4^{2})$$(54)$$(5^{2})$$(65)$$(6^{2})$TTTLegend:$\mathrm{dl}=2$:$\triangle$$\zeta=1^{2},\sigma=2$$\zeta=1^{2},\sigma=1$$\zeta=1^{2},\sigma=0$$\zeta=1,\sigma=2$$\zeta=1,\sigma=1$$\zeta=1,\sigma=0$$\mathrm{dl}=3$:
$\triangle$
$\zeta=1^{2},\sigma=2$$\zeta=1^{2},\sigma=1$$\zeta=1^{2},\sigma=0$$\zeta=1,\sigma=2$$\zeta=1,\sigma=1$$\zeta=1,\sigma=0$depth
$1$pre-periodperiod length $2$ $\triangle$$\triangle$
$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$\ast 2$$\ast 2$$\ast 2$$\ast
7$$\ast 7$$\ast 7$$\ast 5$$\ast 5$$\ast 5$$\ast 3$$\ast 3$$\ast 6$$\ast
6$$\ast 6$$\ast 9$$\ast 9$$\ast 2$$\ast 7$$\ast 9$$\ast 9$$\ast 9$$\ast
16$$\ast 16$infinitemainline$\mathcal{T}^{4}(\langle
2187,64\rangle-\\#2;33-\\#2;25)$$\langle
2187,64\rangle-\\#2;33-\\#2;25$$=P_{11}$branch$\mathcal{B}(11)$$-\\#1;5|6$$4$$2|3$$1$$7|8$$9..15$$18..20$$21..23$$16|24$$17|25..30$$\mathcal{B}(12)$$-\\#1;6$$5$$4$$3$$1|2|15..17$$7..12$$13|14|18..24$$\mathcal{B}(13)$$-\\#1;7|8$$6$$4|5$$3$$1|2$$9..15$$19..27$$16..18|28..40$$\mathcal{B}(14)$$-\\#1;6$$5$$4$$3$$1|2|15..17$$7..12$$13|14|18..24$$\mathcal{B}(15)$$-\\#1;5|6$$4$$2|3$$1$$7|8$$9..15$$22..30$$16..21|31..40$$\mathcal{B}(16)$$-\\#1;4$$3$$2$$1$$5|6|15..17$$9..14$$7|8|18..24$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&0\\\
0&0\end{pmatrix}$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}1&0\\\ \pm
1&0\end{pmatrix}$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}1&0\\\
0&0\end{pmatrix}$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&0\\\ \pm
1&0\end{pmatrix}$$\stackbin[\pm 1]{n-4,n}{G}\begin{pmatrix}\alpha&0\\\
\gamma&0\end{pmatrix}$$\stackbin[\pm 1]{n-4,n}{G}\begin{pmatrix}\alpha&0\\\
\gamma&0\end{pmatrix}$TKTb.10∗d.19d.23d.25b.10b.10b.10b.10b.10$\varkappa=$$(0043)$$(3043)$$(1043)$$(2043)$$(0043)$$(0043)$$(0043)$$(0043)$$(0043)$

######  Theorem 12.2.

(Graph theoretic and algebraic invariants.)
The coclass-$5$ tree $\mathcal{T}:=\mathcal{T}^{5}{P_{11}}$ of finite
$3$-groups $G$ with coclass $\mathrm{cc}(G)=5$ which arises from the
metabelian root $P_{11}:=\langle 2187,64\rangle-\\#2;33-\\#2;25$ has the
following graph theoretic properties.

  1. (1)

The pre-period $(\mathcal{B}(11))$ of length $\ell_{\ast}=1$ is irregular.

  2. (2)

The cardinality of the irregular branch is $\\#\mathcal{B}(11)=30$.

  3. (3)

The branches $\mathcal{B}(i)$, $i\geq p_{\ast}=12$, are periodic with
primitive period $(\mathcal{B}(12),\mathcal{B}(13))$ of length $\ell=2$.

  4. (4)

The cardinalities of the regular branches are $\\#\mathcal{B}(12)=24$ and
$\\#\mathcal{B}(13)=40$.

  5. (5)

Depth, width, and information content of the tree are given by

(12.2)
$\mathrm{dp}(\mathcal{T}^{5}{P_{11}})=1,\quad\mathrm{wd}(\mathcal{T}^{5}{P_{11}})=40,\quad\text{
and }\quad\mathrm{IC}(\mathcal{T}^{5}{P_{11}})=94.$

The algebraic invariants of the groups represented by vertices forming the
pre-period $(\mathcal{B}(11))$ and the primitive period
$(\mathcal{B}(12),\mathcal{B}(13))$ of the tree are given in Table 7. The six
leading branches $\mathcal{B}(11),\ldots,\mathcal{B}(16)$ are drawn in Figure
13.

Table 7. Data for $3$-groups $G$ with $11\leq n=\mathrm{lo}(G)\leq 14$ of the coclass tree $\mathcal{T}^{5}{P_{11}}$ $\\#$ | $m,n$ | $\rho;\alpha,\beta,\gamma,\delta$ | dp | dl | $\zeta$ | $\mu$ | $\nu$ | $\tau(1)$ | $\tau_{2}$ | Type | $\varkappa$ | $\sigma$ | $\\#\mathrm{Aut}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$1$ | $7,11$ | $0;0,0,0,0$ | 0 | 2 | $1^{2}$ | $6$ | $2$ | $3^{2}$ | $32^{3}$ | $\mathrm{b}.10^{\ast}$ | $(0043)$ | $2^{\ast}$ | $2^{3}\cdot 3^{18}$
$1$ | $8,12$ | $0;0,0,0,0$ | 0 | 2 | $1^{2}$ | $6$ | $1$ | $43$ | $3^{2}2^{2}$ | $\mathrm{b}.10^{\ast}$ | $(0043)$ | $2$ | $2^{2}\cdot 3^{20}$
$2$ | $8,12$ | $0;1,0,\pm 1,0$ | 1 | 2 | $1^{2}$ | $5$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{d}.19$ | $(3043)$ | $1^{\ast}$ | $2\cdot 3^{20}$
$1$ | $8,12$ | $0;1,0,0,0$ | 1 | 2 | $1^{2}$ | $5$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{d}.23$ | $(1043)$ | $1^{\ast}$ | $2\cdot 3^{20}$
$2$ | $8,12$ | $0;0,0,\pm 1,0$ | 1 | 2 | $1^{2}$ | $5$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{d}.25$ | $(2043)$ | $2^{\ast}$ | $2^{2}\cdot 3^{20}$
$3$ | $8,12$ | $\pm 1;\alpha,0,\gamma,0$ | 1 | 2 | $1$ | $5$ | $0$ | $3^{2}$ | $3^{2}2^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $2^{\ast}$ | $2^{2}\cdot 3^{20}$
$3$ | $8,12$ | $\pm 1;\alpha,0,\gamma,0$ | 1 | 2 | $1$ | $5$ | $0$ | $3^{2}$ | $3^{2}2^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $1^{\ast}$ | $2\cdot 3^{20}$
$6$ | $8,12$ |  | 1 | 3 | $1^{2}$ | $5$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{b}.10$ | $(0043)$ | $1^{\ast}$ | $2\cdot 3^{19}$
$2$ | $8,12$ |  | 1 | 3 | $1^{2}$ | $5$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{b}.10$ | $(0043)$ | $2^{\ast}$ | $2^{2}\cdot 3^{18}$
$1$ | $8,12$ |  | 1 | 3 | $1^{2}$ | $5$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{b}.10$ | $(0043)$ | $1^{\ast}$ | $2\cdot 3^{18}$
$2$ | $8,12$ |  | 1 | 3 | $1$ | $5$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{b}.10$ | $(0043)$ | $2^{\ast}$ | $2^{2}\cdot 3^{19}$
$5$ | $8,12$ |  | 1 | 3 | $1$ | $5$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{b}.10$ | $(0043)$ | $1^{\ast}$ | $2\cdot 3^{19}$
$2$ | $8,12$ |  | 1 | 3 | $1$ | $5$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{b}.10$ | $(0043)$ | $1^{\ast}$ | $2\cdot 3^{18}$
$1$ | $9,13$ | $0;0,0,0,0$ | 0 | 2 | $1^{2}$ | $6$ | $1$ | $4^{2}$ | $432^{2}$ | $\mathrm{b}.10^{\ast}$ | $(0043)$ | $2^{\ast}$ | $2^{2}\cdot 3^{22}$
$1$ | $9,13$ | $0;1,0,1,0$ | 1 | 2 | $1^{2}$ | $5$ | $0$ | $4^{2}$ | $432^{2}$ | $\mathrm{d}.19$ | $(3043)$ | $0$ | $3^{22}$
$1$ | $9,13$ | $0;1,0,0,0$ | 1 | 2 | $1^{2}$ | $5$ | $0$ | $4^{2}$ | $432^{2}$ | $\mathrm{d}.23$ | $(1043)$ | $0$ | $2\cdot 3^{22}$
$1$ | $9,13$ | $0;0,0,1,0$ | 1 | 2 | $1^{2}$ | $5$ | $0$ | $4^{2}$ | $432^{2}$ | $\mathrm{d}.25$ | $(2043)$ | $0$ | $2\cdot 3^{22}$
$3$ | $9,13$ | $\pm 1;\alpha,0,\gamma,0$ | 1 | 2 | $1$ | $5$ | $0$ | $43$ | $432^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $0$ | $2\cdot 3^{22}$
$3$ | $9,13$ | $\pm 1;\alpha,0,\gamma,0$ | 1 | 2 | $1$ | $5$ | $0$ | $43$ | $432^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $0$ | $3^{22}$
$3$ | $9,13$ |  | 1 | 3 | $1^{2}$ | $5$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $0$ | $3^{21}$
$2$ | $9,13$ |  | 1 | 3 | $1^{2}$ | $5$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $0$ | $2\cdot 3^{20}$
$6$ | $9,13$ |  | 1 | 3 | $1$ | $5$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $0$ | $3^{21}$
$2$ | $9,13$ |  | 1 | 3 | $1$ | $5$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $0$ | $2\cdot 3^{20}$
$1$ | $9,13$ |  | 1 | 3 | $1$ | $5$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $0$ | $3^{20}$
$1$ | $10,14$ | $0;0,0,0,0$ | 0 | 2 | $1^{2}$ | $6$ | $1$ | $54$ | $4^{2}2^{2}$ | $\mathrm{b}.10^{\ast}$ | $(0043)$ | $2$ | $2^{2}\cdot 3^{24}$
$2$ | $10,14$ | $0;1,0,\pm 1,0$ | 1 | 2 | $1^{2}$ | $5$ | $0$ | $54$ | $4^{2}2^{2}$ | $\mathrm{d}.19$ | $(3043)$ | $1^{\ast}$ | $2\cdot 3^{24}$
$1$ | $10,14$ | $0;1,0,0,0$ | 1 | 2 | $1^{2}$ | $5$ | $0$ | $54$ | $4^{2}2^{2}$ | $\mathrm{d}.23$ | $(1043)$ | $1^{\ast}$ | $2\cdot 3^{24}$
$2$ | $10,14$ | $0;0,0,\pm 1,0$ | 1 | 2 | $1^{2}$ | $5$ | $0$ | $54$ | $4^{2}2^{2}$ | $\mathrm{d}.25$ | $(2043)$ | $2^{\ast}$ | $2^{2}\cdot 3^{24}$
$9$ | $10,14$ | $\pm 1;\alpha,0,\gamma,0$ | 1 | 2 | $1$ | $5$ | $0$ | $4^{2}$ | $4^{2}2^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $1^{\ast}$ | $2\cdot 3^{24}$
$6$ | $10,14$ |  | 1 | 3 | $1^{2}$ | $5$ | $0$ | $4^{2}$ | $432^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $1^{\ast}$ | $2\cdot 3^{23}$
$2$ | $10,14$ |  | 1 | 3 | $1^{2}$ | $5$ | $0$ | $4^{2}$ | $432^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $2^{\ast}$ | $2^{2}\cdot 3^{22}$
$1$ | $10,14$ |  | 1 | 3 | $1^{2}$ | $5$ | $0$ | $4^{2}$ | $432^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $1^{\ast}$ | $2\cdot 3^{22}$
$12$ | $10,14$ |  | 1 | 3 | $1$ | $5$ | $0$ | $4^{2}$ | $432^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $1^{\ast}$ | $2\cdot 3^{23}$
$4$ | $10,14$ |  | 1 | 3 | $1$ | $5$ | $0$ | $4^{2}$ | $432^{2}$ | $\mathrm{b}.10$ | $(0043)$ | $1^{\ast}$ | $2\cdot 3^{22}$

######  Remark 12.1.

The algebraic information in Table 7 is visualized in Figure 13. By periodic
continuation, the figure shows more branches than the table but less details
concerning the exact order $\\#\mathrm{Aut}$ of the automorphism group.

###### Proof.

(of Theorem 12.2) According to Proposition 12.2, the logarithmic order of the
tree root, respectively of the periodic root, is $n_{\ast}=11$, respectively
$p_{\ast}=n_{\ast}+\ell_{\ast}=12$.

Since $C_{1}(m_{n})=1$ for all mainline vertices $m_{n}$ with $n\geq
n_{\ast}$, according to Proposition 12.2, the unique capable child of $m_{n}$
is $m_{n+1}$, and each branch has depth $\mathrm{dp}(\mathcal{B}(n))=1$, for
$n\geq n_{\ast}$. Consequently, the tree is also of depth
$\mathrm{dp}(\mathcal{T})=1$.

With the aid of Formula (5.9) in Theorem 5.1, the claims (2) and (4) are
consequences of Proposition 12.2:
$\\#\mathcal{B}(11)=N_{1}(m_{11})=30$, $\\#\mathcal{B}(12)=N_{1}(m_{12})=24$,
and $\\#\mathcal{B}(13)=N_{1}(m_{13})=40$.

According to Formula (5.13) in Corollary 5.1, where $n$ runs from
$n_{\ast}+1=12$ to $n_{\ast}+\ell_{\ast}+\ell+0=14$, the tree width is the
maximum $\mathrm{wd}(\mathcal{T})=40$ of the expressions $N_{1}(m_{11})=30$,
$N_{1}(m_{12})=24$, and $N_{1}(m_{13})=40$.

The information content of the tree is given by Formula (5.17) in the
Definition 5.3:
$\mathrm{IC}(\mathcal{T})=\\#\mathcal{B}(11)+(\\#\mathcal{B}(12)+\\#\mathcal{B}(13))=30+(24+40)=94$.

The algebraic invariants in Table 7, that is, depth $\mathrm{dp}$, derived
length $\mathrm{dl}$, abelian type invariants of the centre $\zeta$, relation
rank $\mu$, nuclear rank $\nu$, abelian quotient invariants $\tau(1)$ of the
first maximal subgroup, respectively $\tau_{2}$ of the commutator subgroup,
transfer kernel type $\varkappa$, action flag $\sigma$, and the factorized
order $\\#\mathrm{Aut}$ of the automorphism group have been computed by means
of program scripts written for MAGMA [17].

Each group is characterized by the parameters of the normalized representative
$G_{\rho}^{m,n}(\alpha,\beta,\gamma,\delta)$ of its isomorphism class,
according to Formula (9.2), and by its identifier $\langle 3^{n},i\rangle$ in
the SmallGroups Database [19].

The column with header $\\#$ contains the number of groups with identical
invariants (except the presentation), for each row. ∎

######  Corollary 12.1.

(Actions and relation ranks.) The algebraic invariants of the vertices of the
structured coclass-$5$ tree $\mathcal{T}^{5}{P_{11}}$ are listed in Table 7.
In particular:

  1. (1)

The groups with $V_{4}$-action are all mainline vertices
$\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$, $n\geq 11$,
the two terminal vertices $\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&0\\\ \pm
1&0\end{pmatrix}$ with even $n\geq 12$, two terminal non-metabelian vertices
with $\zeta=1^{2}$ and even $n\geq 12$, three pre-periodic terminal vertices
$\stackbin[\pm 1]{8,12}{G}\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$, and two
pre-periodic terminal non-metabelian vertices with $\zeta=1$ and $n=12$.

  2. (2)

With respect to the kernel types, all mainline groups of type
$\mathrm{b}.10^{\ast}$, $\varkappa=(0043)$, the two leaves of type
$\mathrm{d}.25$, $\varkappa\sim(2043)$, with every even logarithmic order, two
distinguished non-metabelian leaves of type $\mathrm{b}.10$ with every even
logarithmic order, and five pre-periodic leaves of type $\mathrm{b}.10$ with
$n=12$ possess a $V_{4}$-action.

  3. (3)

All terminal vertices of depth $1$ with odd class and the mainline vertices
with even class possess an RI-action.

  4. (4)

The relation rank is given by $\mu=6$ for the mainline vertices
$\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&0\\\ 0&0\end{pmatrix}$, $n\geq 11$,
and $\mu=5$ otherwise.

###### Proof.

(of Corollary 12.1) The existence of an RI-action on $G$ has been checked by
means of an algorithm involving the $p$-covering group of $G$, written for
MAGMA [17]. The other claims follow immediately from Table 7, continued
indefinitely with the aid of the periodicity in Proposition 12.2. ∎

Figure 14. The unique coclass-$5$ tree $\mathcal{T}^{5}(P_{9}-\\#2;29)$ with
mainline of type $\mathrm{d}.19^{\ast}$order
$3^{n}$$177\,147$$3^{11}$$531\,441$$3^{12}$$1\,594\,323$$3^{13}$$4\,782\,969$$3^{14}$$14\,348\,907$$3^{15}$$43\,046\,721$$3^{16}$$129\,140\,163$$3^{17}$$\tau(1)=$$(3^{2})$$(43)$$(4^{2})$$(54)$$(5^{2})$$(65)$$(6^{2})$TTTLegend:$\mathrm{dl}=2$:$\zeta=1^{2},\sigma=0$$\zeta=1,\sigma=0$$\mathrm{dl}=3$:$\zeta=1^{2},\sigma=0$$\zeta=1,\sigma=0$depth
$2$period length
$1$$16\ast$$16\ast$$16\ast$$16\ast$$16\ast$$16\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$9\ast$$9\ast$$9\ast$$9\ast$$9\ast$$16\ast$$16\ast$$16\ast$$16\ast$$16\ast$infinitemainline$\mathcal{T}^{5}(\langle
2187,64\rangle-\\#2;33-\\#2;29)$$\langle
2187,64\rangle-\\#2;33-\\#2;29$branch$\mathcal{B}(11)$$-\\#1;8|12$$9|11$$14|15$$1..6|$$16..25$$13$$7|10$$\mathcal{B}(12)$$-\\#1;6|11$$5|12$$8|9$$1..3|$$13..25$$7$$4|10$$7..15$$1..9$$1..6|16..25$$10..25$$\mathcal{B}(13)$$-\\#1;5|9$$6|8$$2|3$$10..25$$1$$4|7$$7..15$$4..12$$1..6|16..25$$1..3|13..25$$\mathcal{B}(14)$$-\\#1;5|9$$6|8$$2|3$$10..25$$1$$4|7$$4..12$$7..15$$1..3|13..25$$1..6|16..25$$\mathcal{B}(15)$$-\\#1;5|9$$6|8$$2|3$$10..25$$1$$4|7$$4..12$$7..15$$1..3|13..25$$1..6|16..25$$\mathcal{B}(16)$$-\\#1;5|9$$6|8$$2|3$$10..25$$1$$4|7$$4..12$$7..15$$1..3|13..25$$1..6|16..25$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&1\\\
0&1\end{pmatrix}$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}\pm 1&1\\\ \mp
1&1\end{pmatrix}$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}\pm 1&1\\\
0&1\end{pmatrix}$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&1\\\ \pm
1&1\end{pmatrix}$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}\pm 1&1\\\ \pm
1&1\end{pmatrix}$$\stackbin[\pm 1]{n-4,n}{G}\begin{pmatrix}\alpha&1\\\
\gamma&1\end{pmatrix}$TKTd.19∗d.19F.7F.12F.13H.4H.4H.4H.4$\varkappa=$$(0343)$$(0343)$$(4343)$$(1343)$$(2343)$$(3343)$$(3343)$$(3343)$$(3343)$

###  12.2. The unique mainline of type $\mathrm{d}.19^{\ast}$ for odd coclass
$r\geq 5$

######  Proposition 12.3.

(Periodicity and descendant numbers.)
The branches $\mathcal{B}(i)$, $i\geq n_{\ast}=11$, of the unique coclass-$5$
tree $\mathcal{T}^{5}(P_{9}-\\#2;29)$ with mainline vertices of transfer
kernel type $\mathrm{d}.19^{\ast}$, $\varkappa\sim(0343)$, are purely periodic
with primitive length $\ell=1$ and without pre-period, $\ell_{\ast}=0$, that
is, $\mathcal{B}(i+1)\simeq\mathcal{B}(i)$ are isomorphic as structured
digraphs, for all $i\geq p_{\ast}=n_{\ast}+\ell_{\ast}=11$.

The graph theoretic structure of the tree is determined uniquely by the
numbers $N_{1}$ of immediate descendants and $C_{1}$ of capable immediate
descendants for mainline vertices $m_{n}$ with logarithmic order
$n=\mathrm{lo}(m_{n})\geq n_{\ast}=11$ and for capable vertices $v$ with depth
$1$ and $\mathrm{lo}(v)\geq n_{\ast}+1=12$:
$(N_{1},C_{1})=(25,3)$ for all mainline vertices $m_{n}$ of any logarithmic
order $n\geq 11$,
$(N_{1},C_{1})=(25,0)$ for two capable vertices $v$ of depth $1$ and any
logarithmic order $\mathrm{lo}(v)\geq 12$.

###### Proof.

(of Proposition 12.3) The statements concerning the numbers $N_{1}(m_{n})$ of
immediate descendants of the mainline vertices $m_{n}$ with $n\geq
n_{\ast}=11$, and $N_{1}(v)$ of vertices with depth $\mathrm{dp}(v)=1$ and
logarithmic order $\mathrm{lo}(v)\geq n_{\ast}+1=12$, have been obtained by
direct computation with the $p$-group generation algorithm [12, 13, 14] in
MAGMA [17]. In detail, we proved that there is no pre-period, $\ell_{\ast}=0$,
and the primitive period $(\mathcal{B}(11))$ of length $\ell=1$ consists of
$9$, resp. $18$, metabelian vertices with $\zeta=1^{2}$, resp. $\zeta=1$, and
$16$, resp. $32$, non-metabelian vertices with $\zeta=1^{2}$, resp. $\zeta=1$,
(i.e. $25=9+16$ children of $m_{11}$, and $50=2\cdot(9+16)$ children of
$v_{2,3}(m_{11})$, both with depth $1$)
together $75$ vertices ($27$ of them metabelian) in branch $\mathcal{B}(11)$.

The results concerning the metabelian skeleton confirm the corresponding
statements in the dissertation of Nebelung [7, Thm. 5.1.16, pp. 178–179, and
the third Figure, $e\geq 4$, $e\equiv 0\pmod{2}$, on the third page between
pp. 191–192]. The tree $\mathcal{T}^{5}{R_{2}^{5}}$ corresponds to the
infinite metabelian pro-$3$ group $S_{4,4}$ in [6, Cnj. 15 (a), p. 116].

The claim of the virtual periodicity of branches has been proved generally for
any coclass tree in [1] and [2]. Here, the strict periodicity was confirmed by
computation up to branch $\mathcal{B}(30)$ and certainly sets in at
$p_{\ast}=11$. ∎

######  Theorem 12.3.

(Graph theoretic and algebraic invariants.)
The coclass-$5$ tree $\mathcal{T}:=\mathcal{T}^{5}{R_{2}^{5}}$ of $3$-groups
$G$ with coclass $\mathrm{cc}(G)=5$ which arises from the metabelian root
$R_{2}^{5}:=P_{9}-\\#2;29$ has the following abstract graph theoretic
properties.

  1. (1)

The branches $\mathcal{B}(i)$, $i\geq 11$, are purely periodic with primitive
period $(\mathcal{B}(11))$ of length $\ell=1$.

  2. (2)

The cardinality of the periodic branch is $\\#\mathcal{B}(11)=75$.

  3. (3)

Depth, width, and information content of the tree are given by

(12.3)
$\mathrm{dp}(\mathcal{T}^{5}{R_{2}^{5}})=2,\quad\mathrm{wd}(\mathcal{T}^{5}{R_{2}^{5}})=75,\quad\text{
and }\quad\mathrm{IC}(\mathcal{T}^{5}{R_{2}^{5}})=75.$

The algebraic invariants of the vertices forming the root and the primitive
period $(\mathcal{B}(11))$ of the tree are presented in Table 8. The leading
six branches $\mathcal{B}(11),\ldots,\mathcal{B}(16)$ are drawn in Figure 14.

Table 8. Data for $3$-groups $G$ with $11\leq n=\mathrm{lo}(G)\leq 13$ of the coclass tree $\mathcal{T}^{5}{R_{2}^{5}}$ $\\#$ | $m,n$ | $\rho;\alpha,\beta,\gamma,\delta$ | dp | dl | $\zeta$ | $\mu$ | $\nu$ | $\tau(1)$ | $\tau_{2}$ | Type | $\varkappa$ | $\sigma$ | $\\#\mathrm{Aut}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$1$ | $7,11$ | $0;0,1,0,1$ | 0 | 2 | $1^{2}$ | $5$ | $1$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.19^{\ast}$ | $(0343)$ | $0$ | $3^{18}$
$1$ | $8,12$ | $0;0,1,0,1$ | 0 | 2 | $1^{2}$ | $5$ | $1$ | $43$ | $3^{2}2^{2}$ | $\mathrm{d}.19^{\ast}$ | $(0343)$ | $0$ | $3^{20}$
$2$ | $8,12$ | $0;\pm 1,1,\mp 1,1$ | 1 | 2 | $1^{2}$ | $4$ | $0$ | $43$ | $32^{2}1$ | $\mathrm{F}.7$ | $(4343)$ | $0$ | $3^{20}$
$2$ | $8,12$ | $0;\pm 1,1,0,1$ | 1 | 2 | $1^{2}$ | $4$ | $0$ | $43$ | $32^{2}1$ | $\mathrm{F}.12$ | $(1343)$ | $0$ | $3^{20}$
$2$ | $8,12$ | $0;0,1,\pm 1,1$ | 1 | 2 | $1^{2}$ | $4$ | $0$ | $43$ | $32^{2}1$ | $\mathrm{F}.13$ | $(2343)$ | $0$ | $3^{20}$
$2$ | $8,12$ | $0;\pm 1,1,\pm 1,1$ | 1 | 2 | $1^{2}$ | $5$ | $1$ | $43$ | $32^{2}1$ | $\mathrm{H}.4$ | $(3343)$ | $0$ | $3^{20}$
$12$ | $8,12$ |  | 1 | 3 | $1^{2}$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.19$ | $(0343)$ | $0$ | $3^{19}$
$4$ | $8,12$ |  | 1 | 3 | $1^{2}$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.19$ | $(0343)$ | $0$ | $3^{18}$
$9$ | $9,13$ | $\pm 1;\alpha,1,\gamma,1$ | 2 | 2 | $1$ | $4$ | $0$ | $43$ | $432^{2}$ | $\mathrm{H}.4$ | $(3343)$ | $0$ | $3^{22}$
$12$ | $9,13$ |  | 2 | 3 | $1$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{H}.4$ | $(3343)$ | $0$ | $3^{21}$
$4$ | $9,13$ |  | 2 | 3 | $1$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{H}.4$ | $(3343)$ | $0$ | $3^{20}$

###### Proof.

(Proof of Theorem 12.3) Since every mainline vertex $m_{n}$ of the tree
$\mathcal{T}$ has three capable children, $C_{1}(m_{n})=3$, but every capable
vertex $v$ of depth $1$ has only terminal children, $C_{1}(v)=0$, according to
Proposition 12.3, the depth of the tree is $\mathrm{dp}(\mathcal{T})=2$. In
this case, the cardinality of a branch $\mathcal{B}$ is the sum of the number
$N_{1}(m_{n})$ of immediate descendants of the branch root $m_{n}$ and the
numbers $N_{1}(v_{i})$ of terminal children of capable vertices $v_{i}$ of
depth $1$ with $2\leq i\leq C_{1}(m_{n})$ (excluding the next mainline vertex
$v_{1}=m_{n+1}$), according to Formula (5.10), that is,

$\\#\mathcal{B}=N_{1}(m_{n})+\sum_{i=2}^{C_{1}(m_{n})}\,N_{1}(v_{i}).$

Applied to the primitive period, this yields $\\#\mathcal{B}(11)=25+25+25=75$.
According to Formula (5.14), the width of the tree is the maximum of all sums
of the shape

$\\#\mathrm{Lyr}_{n}{\mathcal{T}}=\\#\\{v\in\mathcal{T}\mid\mathrm{lo}(v)=n\\}=N_{1}(m_{n-1})+\sum_{i=1}^{C_{1}(m_{n-2})}\,N_{1}(v_{i}(m_{n-2})),$

taken over all branch roots $m_{n}$ with logarithmic orders $n_{\ast}+2\leq
n\leq n_{\ast}+\ell_{\ast}+\ell+1$. Applied to $n_{\ast}=11$, $\ell_{\ast}=0$,
and $\ell=1$, this yields
$\mathrm{wd}(\mathcal{T})=\max(25+25+25)=\max(75)=75$.
Finally, we have $\mathrm{IC}(\mathcal{T})=\\#\mathcal{B}(11)=75$. ∎

######  Corollary 12.2.

(Actions and relation ranks.)
The algebraic invariants of the vertices of the structured coclass-$5$ tree
$\mathcal{T}^{5}{R_{2}^{5}}$ are listed in Table 8. In particular:

  1. (1)

There are no groups with GI-action, let alone with RI- or $V_{4}$-action.

  2. (2)

The relation rank is given by $\mu=5$ for the mainline vertices
$\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&1\\\ 0&1\end{pmatrix}$ with $n\geq
11$, and the capable vertices $\stackbin[0]{n-4,n}{G}\begin{pmatrix}\pm 1&1\\\
\pm 1&1\end{pmatrix}$ of depth $1$ with $n\geq 12$, and $\mu=4$ otherwise.

###### Proof.

(of Corollary 12.2) The existence of an RI-action on $G$ has been checked by
means of an algorithm involving the $p$-covering group of $G$, written for
MAGMA [17]. The other claims follow immediately from Table 8, continued
indefinitely with the aid of the periodicity in Proposition 12.3. ∎

Figure 15. The unique coclass-$5$ tree $\mathcal{T}^{5}(P_{9}-\\#2;37)$ with
mainline of type $\mathrm{d}.23^{\ast}$order
$3^{n}$$177\,147$$3^{11}$$531\,441$$3^{12}$$1\,594\,323$$3^{13}$$4\,782\,969$$3^{14}$$14\,348\,907$$3^{15}$$43\,046\,721$$3^{16}$$129\,140\,163$$3^{17}$$\tau(1)=$$(3^{2})$$(43)$$(4^{2})$$(54)$$(5^{2})$$(65)$$(6^{2})$TTTLegend:$\mathrm{dl}=2$:$\zeta=1^{2},\sigma=0$$\zeta=1,\sigma=0$$\mathrm{dl}=3$:$\zeta=1^{2},\sigma=0$$\zeta=1,\sigma=0$depth
$2$period length
$1$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$9\ast$$9\ast$$9\ast$$9\ast$$9\ast$$9\ast$$9\ast$$9\ast$$9\ast$$9\ast$$9\ast$$16\ast$$16\ast$$16\ast$$16\ast$$16\ast$infinitemainline$\mathcal{T}^{5}(\langle
2187,64\rangle-\\#2;33-\\#2;37)$$\langle
2187,64\rangle-\\#2;33-\\#2;37$branch$\mathcal{B}(11)$$-\\#1;5|7$$3|6$$1|2|$$9..15$$8$$4$$\mathcal{B}(12)$$-\\#1;3|6$$2|7$$1|$$8..15$$5$$4$$1..9$$10..25$$\mathcal{B}(13)$$-\\#1;2|3$$4|6$$7..15$$1$$5$$1..9$$10..25$$\mathcal{B}(14)$$-\\#1;2|3$$4|6$$7..15$$1$$5$$1..9$$10..25$$\mathcal{B}(15)$$-\\#1;2|3$$4|6$$7..15$$1$$5$$1..9$$10..25$$\mathcal{B}(16)$$-\\#1;2|3$$4|6$$7..15$$1$$5$$1..9$$10..25$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&0\\\
0&1\end{pmatrix}$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&0\\\ \pm
1&1\end{pmatrix}$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}1&0\\\ \pm
1&1\end{pmatrix}$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}$$\stackbin[\pm 1]{n-4,n}{G}\begin{pmatrix}\alpha&0\\\
\gamma&1\end{pmatrix}$TKTd.23∗d.23F.11F.12G.16G.16G.16G.16$\varkappa=$$(0243)$$(0243)$$(2243)$$(3243)$$(1243)$$(1243)$$(1243)$$(1243)$

###  12.3. The unique mainline of type $\mathrm{d}.23^{\ast}$ for odd coclass
$r\geq 5$

######  Proposition 12.4.

(Periodicity and descendant numbers.)
The branches $\mathcal{B}(i)$, $i\geq n_{\ast}=11$, of the unique coclass-$5$
tree $\mathcal{T}^{5}(P_{9}-\\#2;37)$ with mainline vertices of transfer
kernel type $\mathrm{d}.23^{\ast}$, $\varkappa\sim(0243)$, are purely periodic
with primitive length $\ell=1$ and without pre-period, $\ell_{\ast}=0$, that
is, $\mathcal{B}(i+1)\simeq\mathcal{B}(i)$ are isomorphic as structured
digraphs, for all $i\geq p_{\ast}=n_{\ast}+\ell_{\ast}=11$.

The graph theoretic structure of the tree is determined uniquely by the
numbers $N_{1}$ of immediate descendants and $C_{1}$ of capable immediate
descendants for mainline vertices $m_{n}$ with logarithmic order
$n=\mathrm{lo}(m_{n})\geq n_{\ast}=11$ and for capable vertices $v$ with depth
$1$ and $\mathrm{lo}(v)\geq n_{\ast}+1=12$:
$(N_{1},C_{1})=(15,2)$ for all mainline vertices $m_{n}$ of any logarithmic
order $n\geq 11$,
$(N_{1},C_{1})=(25,0)$ for the capable vertex $v$ of depth $1$ and any
logarithmic order $\mathrm{lo}(v)\geq 12$.

###### Proof.

(of Proposition 12.4) The statements concerning the numbers $N_{1}(m_{n})$ of
immediate descendants of the mainline vertices $m_{n}$ with $n\geq
n_{\ast}=11$, and $N_{1}(v)$ of vertices with depth $\mathrm{dp}(v)=1$ and
logarithmic order $\mathrm{lo}(v)\geq n_{\ast}+1=12$, have been obtained by
direct computation with the $p$-group generation algorithm [12, 13, 14] in
MAGMA [17]. In detail, we proved that there is no pre-period, $\ell_{\ast}=0$,
and the primitive period $(\mathcal{B}(11))$ of length $\ell=1$ consists of
$6$, resp. $9$, metabelian vertices with $\zeta=1^{2}$, resp. $\zeta=1$, and
$9$, resp. $16$, non-metabelian vertices with $\zeta=1^{2}$, resp. $\zeta=1$,
(i.e. $15=6+9$ children of $m_{11}$, and $25=9+16)$ children of
$v_{2}(m_{11})$ with depth $1$)
together $40$ vertices ($15$ of them metabelian) in branch $\mathcal{B}(11)$.

The results concerning the metabelian skeleton confirm the corresponding
statements in the dissertation of Nebelung [7, Thm. 5.1.16, pp. 178–179, and
the third Figure, $e\geq 4$, $e\equiv 0\pmod{2}$, on the third page between
pp. 191–192]. The tree $\mathcal{T}^{5}{R_{3}^{5}}$ corresponds to the
infinite metabelian pro-$3$ group $S_{4,2}$ in [6, Cnj. 15 (a), p. 116].

The claim of the virtual periodicity of branches has been proved generally for
any coclass tree in [1] and [2]. Here, the strict periodicity was confirmed by
computation up to branch $\mathcal{B}(30)$ and certainly sets in at
$p_{\ast}=11$. ∎

######  Theorem 12.4.

(Graph theoretic and algebraic invariants.)
The coclass-$5$ tree $\mathcal{T}:=\mathcal{T}^{5}{R_{3}^{5}}$ of $3$-groups
$G$ with coclass $\mathrm{cc}(G)=5$ arises from the metabelian root
$R_{3}^{5}:=P_{9}-\\#2;37$ and has the following abstract graph theoretic
properties.

  1. (1)

The branches $\mathcal{B}(i)$, $i\geq 11$, are purely periodic with primitive
period $(\mathcal{B}(11))$ of length $\ell=1$.

  2. (2)

The cardinality of the periodic branch is $\\#\mathcal{B}(11)=40$.

  3. (3)

Depth, width, and information content of the tree are given by

(12.4)
$\mathrm{dp}(\mathcal{T}^{5}{R_{3}^{5}})=2,\quad\mathrm{wd}(\mathcal{T}^{5}{R_{3}^{5}})=40,\quad\text{
and }\quad\mathrm{IC}(\mathcal{T}^{5}{R_{3}^{5}})=40.$

The algebraic invariants of the vertices forming the root and the primitive
period $(\mathcal{B}(11))$ of the tree are presented in Table 9. The leading
six branches $\mathcal{B}(11),\ldots,\mathcal{B}(16)$ are drawn in Figure 15.

Table 9. Data for $3$-groups $G$ with $11\leq n=\mathrm{lo}(G)\leq 13$ of the coclass tree $\mathcal{T}^{5}{R_{3}^{5}}$ $\\#$ | $m,n$ | $\rho;\alpha,\beta,\gamma,\delta$ | dp | dl | $\zeta$ | $\mu$ | $\nu$ | $\tau(1)$ | $\tau_{2}$ | Type | $\varkappa$ | $\sigma$ | $\\#\mathrm{Aut}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$1$ | $7,11$ | $0;0,0,0,1$ | 0 | 2 | $1^{2}$ | $5$ | $1$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.23^{\ast}$ | $(0243)$ | $0$ | $3^{18}$
$1$ | $8,12$ | $0;0,0,0,1$ | 0 | 2 | $1^{2}$ | $5$ | $1$ | $43$ | $3^{2}2^{2}$ | $\mathrm{d}.23^{\ast}$ | $(0243)$ | $0$ | $3^{20}$
$2$ | $8,12$ | $0;0,0,\pm 1,1$ | 1 | 2 | $1^{2}$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{F}.11$ | $(2243)$ | $0$ | $2\cdot 3^{20}$
$2$ | $8,12$ | $0;1,0,\pm 1,1$ | 1 | 2 | $1^{2}$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{F}.12$ | $(3243)$ | $0$ | $3^{20}$
$1$ | $8,12$ | $0;1,0,0,1$ | 1 | 2 | $1^{2}$ | $5$ | $1$ | $43$ | $3^{2}2^{2}$ | $\mathrm{G}.16$ | $(1243)$ | $0$ | $3^{20}$
$6$ | $8,12$ |  | 1 | 3 | $1^{2}$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.23$ | $(0243)$ | $0$ | $3^{19}$
$2$ | $8,12$ |  | 1 | 3 | $1^{2}$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.23$ | $(0243)$ | $0$ | $2\cdot 3^{18}$
$1$ | $8,12$ |  | 1 | 3 | $1^{2}$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.23$ | $(0243)$ | $0$ | $3^{18}$
$9$ | $9,13$ | $\pm 1;\alpha,0,\gamma,1$ | 2 | 2 | $1$ | $4$ | $0$ | $43$ | $432^{2}$ | $\mathrm{G}.16$ | $(1243)$ | $0$ | $3^{22}$
$12$ | $9,13$ |  | 2 | 3 | $1$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{G}.16$ | $(1243)$ | $0$ | $3^{21}$
$4$ | $9,13$ |  | 2 | 3 | $1$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{G}.16$ | $(1243)$ | $0$ | $3^{20}$

###### Proof.

(Proof of Theorem 12.4) Since every mainline vertex $m_{n}$ of the tree
$\mathcal{T}$ has two capable children, $C_{1}(m_{n})=2$, but every capable
vertex $v$ of depth $1$ has only terminal children, $C_{1}(v)=0$, according to
Proposition 12.4, the depth of the tree is $\mathrm{dp}(\mathcal{T})=2$. In
this case, the cardinality of a branch $\mathcal{B}$ is the sum of the number
$N_{1}(m_{n})$ of immediate descendants of the branch root $m_{n}$ and the
numbers $N_{1}(v_{i})$ of terminal children of capable vertices $v_{i}$ of
depth $1$, $2\leq i\leq C_{1}(m_{n})$ ($v_{1}$ the next mainline vertex must
be omitted), according to Formula (5.10),

$\\#\mathcal{B}=N_{1}(m_{n})+\sum_{i=2}^{C_{1}(m_{n})}\,N_{1}(v_{i}).$

Applied to the primitive period, this yields $\\#\mathcal{B}(11)=15+25=40$.
According to Formula (5.14), the width of the tree is the maximum of all sums
of the shape

$\\#\mathrm{Lyr}_{n}{\mathcal{T}}=\\#\\{v\in\mathcal{T}\mid\mathrm{lo}(v)=n\\}=N_{1}(m_{n-1})+\sum_{i=1}^{C_{1}(m_{n-2})}\,N_{1}(v_{i}(m_{n-2})),$

taken over all branch roots $m_{n}$ with logarithmic orders $n_{\ast}+2\leq
n\leq n_{\ast}+\ell_{\ast}+\ell+1$. Applied to $n_{\ast}=11$, $\ell_{\ast}=0$,
and $\ell=1$, this yields $\mathrm{wd}(\mathcal{T})=\max(15+25)=\max(40)=40$.
Finally, we have $\mathrm{IC}(\mathcal{T})=\\#\mathcal{B}(11)=40$. ∎

######  Corollary 12.3.

(Actions and relation ranks.) The algebraic invariants of the vertices of the
structured coclass-$5$ tree $\mathcal{T}^{5}{R_{3}^{5}}$ are listed in Table
9. In particular:

  1. (1)

There are no groups with GI-action, let alone with RI- or $V_{4}$-action.

  2. (2)

The relation rank is given by $\mu=5$ for the mainline vertices
$\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}$ with $n\geq
11$, and the capable vertices $\stackbin[0]{n-4,n}{G}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}$ of depth $1$ with $n\geq 12$, and $\mu=4$ otherwise.

###### Proof.

(of Corollary 12.3) The existence of an RI-action on $G$ has been checked by
means of an algorithm involving the $p$-covering group of $G$, written for
MAGMA [17]. The other claims follow immediately from Table 9, continued
indefinitely with the aid of the periodicity in Prop. 12.4. ∎

Figure 16. The unique coclass-$5$ tree $\mathcal{T}^{5}(P_{9}-\\#2;39)$ with
mainline of type $\mathrm{d}.25^{\ast}$order
$3^{n}$$177\,147$$3^{11}$$531\,441$$3^{12}$$1\,594\,323$$3^{13}$$4\,782\,969$$3^{14}$$14\,348\,907$$3^{15}$$43\,046\,721$$3^{16}$$129\,140\,163$$3^{17}$$\tau(1)=$$(3^{2})$$(43)$$(4^{2})$$(54)$$(5^{2})$$(65)$$(6^{2})$TTTLegend:$\mathrm{dl}=2$:$\zeta=1^{2},\sigma=0$$\zeta=1,\sigma=0$$\mathrm{dl}=3$:$\zeta=1^{2},\sigma=0$$\zeta=1,\sigma=0$depth
$2$period length
$2$$9\ast$$9\ast$$9\ast$$9\ast$$9\ast$$9\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$2\ast$$6\ast$$6\ast$$6\ast$$9\ast$$9\ast$$9\ast$$9\ast$$9\ast$$16\ast$$16\ast$infinitemainline$\mathcal{T}^{5}(\langle
2187,64\rangle-\\#2;33-\\#2;39)$$\langle
2187,64\rangle-\\#2;33-\\#2;39$branch$\mathcal{B}(11)$$-\\#1;2|3$$4$$7..15$$1$$5|6$$\mathcal{B}(12)$$-\\#1;3|6$$2|7$$1|$$8..15$$5$$4$$1..6$$3..8$$7..15$$1|2|9..15$$\mathcal{B}(13)$$-\\#1;5|7$$3$$1|2$$9..15$$8$$4|6$$1..9$$10..25$$\mathcal{B}(14)$$-\\#1;3|6$$2|7$$1|$$8..15$$5$$4$$1..6$$3..8$$7..15$$1|2|9..15$$\mathcal{B}(15)$$-\\#1;2|3$$4$$7..15$$1$$5|6$$1..9$$10..25$$\mathcal{B}(16)$$-\\#1;3|6$$2|7$$1|$$8..15$$5$$4$$1..6$$3..8$$7..15$$1|2|9..15$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&1\\\
0&0\end{pmatrix}$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}\pm 1&1\\\
0&0\end{pmatrix}$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}\pm 1&1\\\
1&0\end{pmatrix}$$\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&1\\\ \pm
1&0\end{pmatrix}$$\stackbin[\pm 1]{n-4,n}{G}\begin{pmatrix}\alpha&1\\\
\gamma&0\end{pmatrix}$TKTd.25∗d.25F.11F.13G.19G.19G.19G.19$\varkappa=$$(0143)$$(0143)$$(1143)$$(3143)$$(2143)$$(2143)$$(2143)$$(2143)$

###  12.4. The unique mainline of type $\mathrm{d}.25^{\ast}$ for odd coclass
$r\geq 5$

######  Proposition 12.5.

(Periodicity and descendant numbers.)
The branches $\mathcal{B}(i)$, $i\geq n_{\ast}=11$, of the unique coclass-$5$
tree $\mathcal{T}^{5}(P_{9}-\\#2;39)$ with mainline vertices of transfer
kernel type $\mathrm{d}.25^{\ast}$, $\varkappa\sim(0143)$, are purely periodic
with primitive length $\ell=2$ and without pre-period, $\ell_{\ast}=0$, that
is, $\mathcal{B}(i+2)\simeq\mathcal{B}(i)$ are isomorphic as structured
digraphs, for all $i\geq p_{\ast}=n_{\ast}+\ell_{\ast}=11$.

The graph theoretic structure of the tree is determined uniquely by the
numbers $N_{1}$ of immediate descendants and $C_{1}$ of capable immediate
descendants for mainline vertices $m_{n}$ with logarithmic order
$n=\mathrm{lo}(m_{n})\geq n_{\ast}=11$ and for capable vertices $v$ with depth
$1$ and $\mathrm{lo}(v)\geq n_{\ast}+1=12$:
$(N_{1},C_{1})=(15,3)$ for all mainline vertices $m_{n}$ of odd logarithmic
order $n\geq 11$,
$(N_{1},C_{1})=(15,2)$ for all mainline vertices $m_{n}$ of even logarithmic
order $n\geq 12$,
$(N_{1},C_{1})=(15,0)$ for two capable vertices $v$ of depth $1$ and even
logarithmic order $\mathrm{lo}(v)\geq 12$,
$(N_{1},C_{1})=(25,0)$ for the capable vertex $v$ of depth $1$ and odd
logarithmic order $\mathrm{lo}(v)\geq 13$.

###### Proof.

(of Proposition 12.5) The statements concerning the numbers $N_{1}(m_{n})$ of
immediate descendants of the mainline vertices $m_{n}$ with $n\geq
n_{\ast}=11$, and $N_{1}(v)$ of vertices with depth $\mathrm{dp}(v)=1$ and
logarithmic order $\mathrm{lo}(v)\geq n_{\ast}+1=12$, have been obtained by
direct computation with the $p$-group generation algorithm [12, 13, 14] in
MAGMA [17]. In detail, we proved that there is no pre-period, $\ell_{\ast}=0$,
and the primitive period $(\mathcal{B}(11),\mathcal{B}(12))$ of length
$\ell=2$ consists of
$6$, resp. $12$, metabelian vertices with $\zeta=1^{2}$, resp. $\zeta=1$, and
$9$, resp. $18$, non-metabelian vertices with $\zeta=1^{2}$, resp. $\zeta=1$,
(i.e. $15=6+9$ children of $m_{11}$, and $30=2\cdot(6+9)$ children of
$v_{2,3}(m_{11})$, both with depth $1$)
together $45$ vertices ($18$ of them metabelian) in branch $\mathcal{B}(11)$,
and
$6$, resp. $9$, metabelian vertices with $\zeta=1^{2}$, resp. $\zeta=1$, and
$9$, resp. $16$, non-metabelian vertices with $\zeta=1^{2}$, resp. $\zeta=1$,
(i.e. $15=6+9$ children of $m_{12}$, and $25=9+16$ children of $v_{2}(m_{12})$
with depth $1$)
together $40$ vertices ($15$ of them metabelian) in branch $\mathcal{B}(12)$.

The results concerning the metabelian skeleton confirm the corresponding
statements in the dissertation of Nebelung [7, Thm. 5.1.16, pp. 178–179, and
the third Figure, $e\geq 4$, $e\equiv 0\pmod{2}$, on the third page between
pp. 191–192]. The tree $\mathcal{T}^{5}{R_{4}^{5}}$ corresponds to the
infinite metabelian pro-$3$ group $S_{4,3}$ in [6, Cnj. 15 (a), p. 116].

The claim of the virtual periodicity of branches has been proved generally for
any coclass tree in [1] and [2]. Here, the strict periodicity was confirmed by
computation up to branch $\mathcal{B}(30)$ and certainly sets in at
$p_{\ast}=11$. ∎

######  Theorem 12.5.

(Graph theoretic and algebraic invariants.)
The coclass-$5$ tree $\mathcal{T}:=\mathcal{T}^{5}{R_{4}^{5}}$ of $3$-groups
$G$ with coclass $\mathrm{cc}(G)=5$ which arises from the metabelian root
$R_{4}^{5}:=P_{9}-\\#2;39$ has the following abstract graph theoretic
properties.

  1. (1)

The branches $\mathcal{B}(i)$, $i\geq 11$, are purely periodic with primitive
period $(\mathcal{B}(11),\mathcal{B}(12))$ of length $\ell=2$.

  2. (2)

The cardinalities of the periodic branches are $\\#\mathcal{B}(11)=45$ and
$\\#\mathcal{B}(12)=40$.

  3. (3)

Depth, width, and information content of the tree are given by

(12.5)
$\mathrm{dp}(\mathcal{T}^{5}{R_{4}^{5}})=2,\quad\mathrm{wd}(\mathcal{T}^{5}{R_{4}^{5}})=45,\quad\text{
and }\quad\mathrm{IC}(\mathcal{T}^{5}{R_{4}^{5}})=85.$

The algebraic invariants of the vertices forming the root and the primitive
period $(\mathcal{B}(11),\mathcal{B}(12))$ of the tree are presented in Table
10. The leading six branches $\mathcal{B}(11),\ldots,\mathcal{B}(16)$ are
drawn in Figure 16.

Table 10. Data for $3$-groups $G$ with $11\leq n=\mathrm{lo}(G)\leq 14$ of the coclass tree $\mathcal{T}^{5}{R_{4}^{5}}$ $\\#$ | $m,n$ | $\rho;\alpha,\beta,\gamma,\delta$ | dp | dl | $\zeta$ | $\mu$ | $\nu$ | $\tau(1)$ | $\tau_{2}$ | Type | $\varkappa$ | $\sigma$ | $\\#\mathrm{Aut}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$1$ | $7,11$ | $0;0,1,0,0$ | 0 | 2 | $1^{2}$ | $5$ | $1$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.25^{\ast}$ | $(0143)$ | $0$ | $2\cdot 3^{18}$
$1$ | $8,12$ | $0;0,1,0,0$ | 0 | 2 | $1^{2}$ | $5$ | $1$ | $43$ | $3^{2}2^{2}$ | $\mathrm{d}.25^{\ast}$ | $(0143)$ | $0$ | $2\cdot 3^{20}$
$1$ | $8,12$ | $0;1,1,0,0$ | 1 | 2 | $1^{2}$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{F}.11$ | $(1143)$ | $0$ | $3^{20}$
$2$ | $8,12$ | $0;1,1,\pm 1,0$ | 1 | 2 | $1^{2}$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{F}.13$ | $(3143)$ | $0$ | $3^{20}$
$2$ | $8,12$ | $0;0,1,\pm 1,0$ | 1 | 2 | $1^{2}$ | $5$ | $1$ | $43$ | $3^{2}2^{2}$ | $\mathrm{G}.19$ | $(2143)$ | $0$ | $2\cdot 3^{20}$
$6$ | $8,12$ |  | 1 | 3 | $1^{2}$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.25$ | $(0143)$ | $0$ | $3^{19}$
$2$ | $8,12$ |  | 1 | 3 | $1^{2}$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.25$ | $(0143)$ | $0$ | $2\cdot 3^{18}$
$1$ | $8,12$ |  | 1 | 3 | $1^{2}$ | $4$ | $0$ | $3^{2}$ | $32^{3}$ | $\mathrm{d}.25$ | $(0143)$ | $0$ | $3^{18}$
$3$ | $9,13$ | $\pm 1;\alpha,1,\gamma,0$ | 2 | 2 | $1$ | $4$ | $0$ | $43$ | $432^{2}$ | $\mathrm{G}.19$ | $(2143)$ | $0$ | $2\cdot 3^{22}$
$3$ | $9,13$ | $\pm 1;\alpha,1,\gamma,0$ | 2 | 2 | $1$ | $4$ | $0$ | $43$ | $432^{2}$ | $\mathrm{G}.19$ | $(2143)$ | $0$ | $3^{22}$
$6$ | $9,13$ |  | 2 | 3 | $1$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{G}.19$ | $(2143)$ | $0$ | $3^{21}$
$2$ | $9,13$ |  | 2 | 3 | $1$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{G}.19$ | $(2143)$ | $0$ | $2\cdot 3^{20}$
$1$ | $9,13$ |  | 2 | 3 | $1$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{G}.19$ | $(2143)$ | $0$ | $3^{20}$
$1$ | $9,13$ | $0;0,1,0,0$ | 0 | 2 | $1^{2}$ | $5$ | $1$ | $4^{2}$ | $432^{2}$ | $\mathrm{d}.25^{\ast}$ | $(0143)$ | $0$ | $2\cdot 3^{22}$
$2$ | $9,13$ | $0;\pm 1,1,0,0$ | 1 | 2 | $1^{2}$ | $4$ | $0$ | $4^{2}$ | $432^{2}$ | $\mathrm{F}.11$ | $(1143)$ | $0$ | $3^{22}$
$2$ | $9,13$ | $0;\pm 1,1,1,0$ | 1 | 2 | $1^{2}$ | $4$ | $0$ | $4^{2}$ | $432^{2}$ | $\mathrm{F}.13$ | $(3143)$ | $0$ | $3^{22}$
$1$ | $9,13$ | $0;0,1,1,0$ | 1 | 2 | $1^{2}$ | $5$ | $1$ | $4^{2}$ | $432^{2}$ | $\mathrm{G}.19$ | $(2143)$ | $0$ | $2\cdot 3^{22}$
$6$ | $9,13$ |  | 1 | 3 | $1^{2}$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{d}.25$ | $(0143)$ | $0$ | $3^{21}$
$2$ | $9,13$ |  | 1 | 3 | $1^{2}$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{d}.25$ | $(0143)$ | $0$ | $2\cdot 3^{20}$
$1$ | $9,13$ |  | 1 | 3 | $1^{2}$ | $4$ | $0$ | $43$ | $3^{2}2^{2}$ | $\mathrm{d}.25$ | $(0143)$ | $0$ | $3^{20}$
$9$ | $10,14$ | $\pm 1;\alpha,1,\gamma,0$ | 2 | 2 | $1$ | $4$ | $0$ | $4^{2}$ | $4^{2}2^{2}$ | $\mathrm{G}.19$ | $(2143)$ | $0$ | $3^{24}$
$12$ | $10,14$ |  | 2 | 3 | $1$ | $4$ | $0$ | $4^{2}$ | $432^{2}$ | $\mathrm{G}.19$ | $(2143)$ | $0$ | $3^{23}$
$4$ | $10,14$ |  | 2 | 3 | $1$ | $4$ | $0$ | $4^{2}$ | $432^{2}$ | $\mathrm{G}.19$ | $(2143)$ | $0$ | $3^{22}$

###### Proof.

(Proof of Theorem 12.5) Since every mainline vertex $m_{n}$ of the tree
$\mathcal{T}$ has several capable children, $C_{1}(m_{n})\geq 2$, but every
capable vertex $v$ of depth $1$ has only terminal children, $C_{1}(v)=0$,
according to Proposition 12.5, the depth of the tree is
$\mathrm{dp}(\mathcal{T})=2$. In this case, the cardinality of a branch
$\mathcal{B}$ is the sum of the number $N_{1}(m_{n})$ of immediate descendants
of the branch root $m_{n}$ and the numbers $N_{1}(v_{i})$ of terminal children
of capable vertices $v_{i}$ of depth $1$, $2\leq i\leq C_{1}(m_{n})$ (where
$v_{1}$ is the next mainline vertex and must be discouraged), according to
Formula (5.10),

$\\#\mathcal{B}=N_{1}(m_{n})+\sum_{i=2}^{C_{1}(m_{n})}\,N_{1}(v_{i}).$

Applied to the primitive period, this yields
$\\#\mathcal{B}(11)=15+(15+15)=45$ and $\\#\mathcal{B}(12)=15+25=40$.
According to Formula (5.14), the width of the tree is the maximum of all sums
of the shape

$\\#\mathrm{Lyr}_{n}{\mathcal{T}}=\\#\\{v\in\mathcal{T}\mid\mathrm{lo}(v)=n\\}=N_{1}(m_{n-1})+\sum_{i=1}^{C_{1}(m_{n-2})}\,N_{1}(v_{i}(m_{n-2})),$

taken over all branch roots $m_{n}$ with logarithmic orders $n_{\ast}+2\leq
n\leq n_{\ast}+\ell_{\ast}+\ell+1$. Applied to $n_{\ast}=11$, $\ell_{\ast}=0$,
and $\ell=2$, this yields
$\mathrm{wd}(\mathcal{T})=\max(15+(15+15),15+25)=\max(45,40)=45$. Finally, we
have $\mathrm{IC}(\mathcal{T})=45+40=85$. ∎

######  Corollary 12.4.

(Actions and relation ranks.)
The algebraic invariants of the vertices of the structured coclass-$5$ tree
$\mathcal{T}^{5}{R_{4}^{5}}$ are listed in Table 10. In particular:

  1. (1)

There are no groups with GI-action, let alone with RI- or $V_{4}$-action.

  2. (2)

The relation rank is given by $\mu=5$ for the mainline vertices
$\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$ with $n\geq
11$, and the capable vertices $\stackbin[0]{n-4,n}{G}\begin{pmatrix}0&1\\\ \pm
1&0\end{pmatrix}$ of depth $1$ with $n\geq 12$, and $\mu=4$ otherwise.

###### Proof.

(of Corollary 12.4) The existence of an RI-action on $G$ has been checked by
means of an algorithm involving the $p$-covering group of $G$, written for
MAGMA [17]. The other claims follow immediately from Table 10, continued
indefinitely with the aid of the periodicity in Proposition 12.5. ∎

##  13\. The forest of $3$-groups with coclass $1$

The coclass forests $\mathcal{F}(1)$ and $\mathcal{F}(2)$, and even their
metabelian skeletons, sporadic parts, and individual coclass trees, do not
reveal any isomorphism to higher coclass forests $\mathcal{F}(r)$, with $r\geq
3$, or parts of them. The metabelian skeleton of the coclass forest
$\mathcal{F}(3)$ is isomorphic to the metabelian skeleton of any coclass
forest $\mathcal{F}(r)$, with odd $r\geq 5$, according to Nebelung [7], but
neither its sporadic part nor its coclass trees are isomorphic to the
corresponding components of other coclass forests.

Whereas the complexity of the pre-periodic forests $\mathcal{F}(2)$ and
$\mathcal{F}(3)$ is very high, the simplest forest $\mathcal{F}(1)$ can be
described easily. In particular, the coclass-$1$ forest
$\mathcal{F}(1)=\mathcal{T}^{1}{R_{1}^{1}}$ coincides with its unique coclass
tree arising from the abelian root $R_{1}^{1}:=\langle 9,2\rangle\simeq
C_{3}\times C_{3}$. Its sporadic part $\mathcal{F}_{0}(1)=\emptyset$ is void.

Figure 17. The unique coclass-$1$ tree $\mathcal{T}^{1}\langle 9,2\rangle$
with mainline of type $\mathrm{a}.1^{\ast}$order
$3^{n}$$9$$3^{2}$$27$$3^{3}$$81$$3^{4}$$243$$3^{5}$$729$$3^{6}$$2\,187$$3^{7}$$6\,561$$3^{8}$$19\,683$$3^{9}$$59\,049$$3^{10}$$\tau(1)=$$(1)$$(1^{2})$$(21)$$(2^{2})$$(32)$$(3^{2})$$(43)$$(4^{2})$$(54)$TTTLegend:$k=0,\sigma=2$$k=0,\sigma=1$$k=0,\sigma=0$$k=1,\sigma=1$$k=1,\sigma=0$depth
$1$pre-periodperiod length $2$infinitemainline$\mathcal{T}^{1}\langle
9,2\rangle$$=C_{3}\times
C_{3}$abelian$=G^{3}_{0}(0,0)$$G^{3}_{0}(0,1)=$$=$$\mathrm{Syl}_{3}A_{9}$TTT$\tau(1)=(1^{3})$$\Phi_{1}$$\langle
2\rangle$$\Phi_{2}$branch$\mathcal{B}(2)$$\langle 4\rangle$$\langle
3\rangle$$\Phi_{3}$$\mathcal{B}(3)$$\langle 8\rangle$$\langle
7\rangle$$\langle 10\rangle$$\langle 9\rangle$$\Phi_{9}$$\langle
25\rangle$$\langle 27\rangle$$\langle
26\rangle$$\Phi_{10}$$\mathcal{B}(4)$$\langle 28\rangle$$\langle
30\rangle$$\langle 29\rangle$$\Phi_{35}$$\langle 98\rangle$$\langle
97\rangle$$\langle 96\rangle$$\langle
95\rangle$$\Phi_{36}$$\mathcal{B}(5)$$\langle 100\rangle$$\langle
99\rangle$$\langle 101\rangle$$\langle 388\rangle$$\langle 387\rangle$$\langle
386\rangle$$\mathcal{B}(6)$$\langle 390\rangle$$\langle 389\rangle$$\langle
391\rangle$$\langle 2224\rangle$$\langle 2223\rangle$$\langle
2222\rangle$$\langle 2221\rangle$$\mathcal{B}(7)$$\langle 2226\rangle$$\langle
2225\rangle$$\langle
2227\rangle$$\\#1;3$$2$$1$$\mathcal{B}(8)$$5$$4$$6$$\mathcal{B}(9)$$G^{n}_{0}(0,0)$$G^{n}_{0}(0,1)$$G^{n}_{0}(1,0)$$G^{n}_{0}(-1,0)$$G^{n}_{1}(0,-1)$$G^{n}_{1}(0,0)$$G^{n}_{1}(0,1)$with
abelian maximal subgroupwithout abelian maximal
subgroupTKTA.1$\varkappa=$$(1111)$TKTa.1∗a.2a.3a.3a.1a.1a.1$\varkappa=$$(0000)$$(1000)$$(2000)$$(2000)$$(0000)$$(0000)$$(0000)$

###  13.1. The unique mainline of type $\mathrm{a}.1^{\ast}$ for coclass $r=1$

######  Proposition 13.1.

(Periodicity and descendant numbers.)
The branches $\mathcal{B}(i)$, $i\geq n_{\ast}=2$, of the coclass-$1$ tree
$\mathcal{T}^{1}\langle 9,2\rangle$ with mainline vertices of transfer kernel
type $\mathrm{a}.1^{\ast}$, $\varkappa=(0000)$, are periodic with pre-period
length $\ell_{\ast}=2$ and with primitive period length $\ell=2$, that is,
$\mathcal{B}(i+2)\simeq\mathcal{B}(i)$ are isomorphic as digraphs, for all
$i\geq p_{\ast}=n_{\ast}+\ell_{\ast}=4$.

The graph theoretic structure of the tree is determined uniquely by the
numbers $N_{1}$ of immediate descendants and $C_{1}$ of capable immediate
descendants of the mainline vertices $m_{n}$ with logarithmic order
$n=\mathrm{lo}(m_{n})\geq n_{\ast}=2$:
$(N_{1},C_{1})=(2,1)$ for the root $m_{2}$ with $n=2$,
$(N_{1},C_{1})=(4,1)$ for the mainline vertex $m_{3}$ with $n=3$,
$(N_{1},C_{1})=(6,1)$ for mainline vertices $m_{n}$ with even logarithmic
order $n\geq 4$,
$(N_{1},C_{1})=(7,1)$ for mainline vertices $m_{n}$ with odd logarithmic order
$n\geq 5$.

###### Proof.

(of Proposition 13.1) The statements concerning the numbers $N_{1}(m_{n})$ of
immediate descendants of the mainline vertices $m_{n}$ with $n\geq 3$ are due
to Blackburn [9, Thm. 4.2 and Thm 4.3, p. 88], who distinguishes the groups
according to their defect of commutativity $k=k(G)\in\\{0,1\\}$, which is
defined by $[\chi_{2}{G},\gamma_{2}{G}]=\gamma_{c+1-k}{G}$ in terms of the
lower central series $(\gamma_{j}{G})_{j\geq 1}$, nilpotency class $c\geq 3$,
and the two-step centralizer $\chi_{2}{G}$ of $G$.

In detail, Blackburn proved that there are
$4$ vertices $v$ with defect $k(v)=0$ in the pre-periodic branch
$\mathcal{B}(3)$,
and the primitive period $(\mathcal{B}(4),\mathcal{B}(5))$ of length $\ell=2$
consists of
$3$ vertices $v$ with defect $k(v)=0$, and $3$ vertices $v$ with defect
$k(v)=1$,
together $6$ vertices in branch $\mathcal{B}(4)$, and
$4$ vertices $v$ with defect $k(v)=0$, and $3$ vertices $v$ with defect
$k(v)=1$,
together $7$ vertices in branch $\mathcal{B}(5)$. All vertices of the tree are
metabelian.

The results were reproduced and supplemented with $N_{1}(m_{2})=2$ by Nebelung
[7, Thm. 5.1.17, pp. 179–180], and have been verified by ourselves
independently by direct computation with MAGMA [17], where the $p$-group
generation algorithm by Newman and O’Brien [12, 13, 14] is implemented.

Accordingly, the pre-period $(\mathcal{B}(2),\mathcal{B}(3))$ of length
$\ell_{\ast}=2$ consists of
$2$ vertices $v$ with defect $k(v)=0$ in branch $\mathcal{B}(2)$, and
$4$ vertices $v$ with defect $k(v)=0$ in branch $\mathcal{B}(3)$.

The claim of the virtual periodicity of branches has been proved generally for
any coclass tree by du Sautoy [1], and independently by Eick and Leedham-Green
[2]. Here, the strict periodicity is also a consequence of Blackburn’s
results, and has been tested up to $\mathcal{B}(25)$ computationally. ∎

Table 11. Data for $3$-groups $G$ with $2\leq n=\mathrm{lo}(G)\leq 6$ of the coclass tree $\mathcal{T}^{1}\langle 9,2\rangle$ $\\#$ | $n$ | $a;z,w$ | $\langle 3^{n},i\rangle$ | $k$ | dp | dl | $\zeta$ | $\mu$ | $\nu$ | $\tau(1)$ | $\tau_{2}$ | Type | $\varkappa$ | $\sigma$ | $\\#\mathrm{Aut}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$1$ | $2$ | $0;0,0$ | $\langle 3^{2},2\rangle$ | $0$ | 0 | 1 | $1^{2}$ | $3$ | $3$ | $1$ | $0$ | $\mathrm{a}.1^{\ast}$ | $(0000)$ | $2$ | $2^{4}\cdot 3$
$1$ | $3$ | $0;0,0$ | $\langle 3^{3},3\rangle$ | $0$ | 0 | 2 | $1$ | $4$ | $2$ | $1^{2}$ | $1$ | $\mathrm{a}.1^{\ast}$ | $(0000)$ | $2^{\ast}$ | $2^{4}\cdot 3^{3}$
$1$ | $3$ | $0;0,1$ | $\langle 3^{3},4\rangle$ | $0$ | 1 | 2 | $1$ | $2$ | $0$ | $1^{2}$ | $1$ | $\mathrm{A}.1$ | $(1111)$ | $0$ | $2\cdot 3^{3}$
$1$ | $4$ | $0;0,0$ | $\langle 3^{4},9\rangle$ | $0$ | 0 | 2 | $1$ | $4$ | $1$ | $21$ | $1^{2}$ | $\mathrm{a}.1^{\ast}$ | $(0000)$ | $2$ | $2^{2}\cdot 3^{5}$
$1$ | $4$ | $0;0,1$ | $\langle 3^{4},10\rangle$ | $0$ | 1 | 2 | $1$ | $3$ | $0$ | $21$ | $1^{2}$ | $\mathrm{a}.2$ | $(1000)$ | $1^{\ast}$ | $2\cdot 3^{5}$
$1$ | $4$ | $0;1,0$ | $\langle 3^{4},7\rangle$ | $0$ | 1 | 2 | $1$ | $3$ | $0$ | $1^{3}$ | $1^{2}$ | $\mathrm{a}.3$ | $(2000)$ | $2^{\ast}$ | $2^{2}\cdot 3^{4}$
$1$ | $4$ | $0;-1,0$ | $\langle 3^{4},8\rangle$ | $0$ | 1 | 2 | $1$ | $3$ | $0$ | $21$ | $1^{2}$ | $\mathrm{a}.3$ | $(2000)$ | $2^{\ast}$ | $2^{2}\cdot 3^{4}$
$1$ | $5$ | $0;0,0$ | $\langle 3^{5},26\rangle$ | $0$ | 0 | 2 | $1$ | $4$ | $1$ | $2^{2}$ | $21$ | $\mathrm{a}.1^{\ast}$ | $(0000)$ | $2^{\ast}$ | $2^{2}\cdot 3^{7}$
$1$ | $5$ | $0;0,1$ | $\langle 3^{5},27\rangle$ | $0$ | 1 | 2 | $1$ | $3$ | $0$ | $2^{2}$ | $21$ | $\mathrm{a}.2$ | $(1000)$ | $0$ | $2\cdot 3^{7}$
$1$ | $5$ | $0;1,0$ | $\langle 3^{5},25\rangle$ | $0$ | 1 | 2 | $1$ | $3$ | $0$ | $2^{2}$ | $21$ | $\mathrm{a}.3$ | $(2000)$ | $0$ | $2\cdot 3^{6}$
$3$ | $5$ | $1;0,w$ | $\langle 3^{5},28..30\rangle$ | $1$ | 1 | 2 | $1$ | $3$ | $0$ | $21$ | $21$ | $\mathrm{a}.1$ | $(0000)$ | $0$ | $2\cdot 3^{6}$
$1$ | $6$ | $0;0,0$ | $\langle 3^{6},95\rangle$ | $0$ | 0 | 2 | $1$ | $4$ | $1$ | $32$ | $2^{2}$ | $\mathrm{a}.1^{\ast}$ | $(0000)$ | $2$ | $2^{2}\cdot 3^{9}$
$1$ | $6$ | $0;0,1$ | $\langle 3^{6},96\rangle$ | $0$ | 1 | 2 | $1$ | $3$ | $0$ | $32$ | $2^{2}$ | $\mathrm{a}.2$ | $(1000)$ | $1^{\ast}$ | $2\cdot 3^{9}$
$2$ | $6$ | $0;\pm 1,0$ | $\langle 3^{6},97|98\rangle$ | $0$ | 1 | 2 | $1$ | $3$ | $0$ | $32$ | $2^{2}$ | $\mathrm{a}.3$ | $(2000)$ | $2^{\ast}$ | $2^{2}\cdot 3^{8}$
$3$ | $6$ | $1;0,w$ | $\langle 3^{6},99..101\rangle$ | $1$ | 1 | 2 | $1$ | $3$ | $0$ | $2^{2}$ | $2^{2}$ | $\mathrm{a}.1$ | $(0000)$ | $1^{\ast}$ | $2\cdot 3^{8}$

######  Theorem 13.1.

(Graph theoretic and algebraic invariants.)
The coclass-$1$ tree $\mathcal{T}:=\mathcal{T}^{1}{R_{1}^{1}}$ of all finite
$3$-groups $G\not\simeq C_{9}$ with coclass $\mathrm{cc}(G)=1$ arises from the
abelian root $R_{1}^{1}:=\langle 9,2\rangle\simeq C_{3}\times C_{3}$ and has
the following graph theoretic properties.

  1. (1)

The pre-period $(\mathcal{B}(2),\mathcal{B}(3))$ of length $\ell_{\ast}=2$ is
irregular.

  2. (2)

The cardinalities of the irregular branches are $\\#\mathcal{B}(2)=2$ and
$\\#\mathcal{B}(3)=4$.

  3. (3)

The branches $\mathcal{B}(i)$, $i\geq 4$, are periodic with primitive period
$(\mathcal{B}(4),\mathcal{B}(5))$ of length $\ell=2$.

  4. (4)

The cardinalities of the regular branches are $\\#\mathcal{B}(4)=6$ and
$\\#\mathcal{B}(5)=7$.

  5. (5)

Depth, width, and information content of the tree are given by

(13.1)
$\mathrm{dp}(\mathcal{T}^{1}{R_{1}^{1}})=1,\quad\mathrm{wd}(\mathcal{T}^{1}{R_{1}^{1}})=7,\quad\text{
and }\quad\mathrm{IC}(\mathcal{T}^{1}{R_{1}^{1}})=19.$

The algebraic invariants of the groups represented by vertices forming the
pre-period $(\mathcal{B}(2),\mathcal{B}(3))$ and the primitive period
$(\mathcal{B}(4),\mathcal{B}(5))$ of the tree are given in Table 11. The
leading eight branches $\mathcal{B}(2),\ldots,\mathcal{B}(9)$ are drawn in
Figure 17. All vertices of the tree are metabelian.

###### Proof.

(of Theorem 13.1) According to Proposition 13.1, the logarithmic order of the
tree root, respectively of the periodic root, is $n_{\ast}=2$, respectively
$p_{\ast}=n_{\ast}+\ell_{\ast}=4$.

Since $C_{1}(m_{n})=1$ for all mainline vertices $m_{n}$ with $n\geq
n_{\ast}$, according to Proposition 13.1, the unique capable child of $m_{n}$
is $m_{n+1}$, and each branch has depth $\mathrm{dp}(\mathcal{B}(n))=1$, for
$n\geq n_{\ast}$. Consequently, the tree is also of depth
$\mathrm{dp}(\mathcal{T})=1$.

With the aid of Formula (5.9) in Theorem 5.1, the claims (2) and (4) are
consequences of Proposition 13.1:
$\\#\mathcal{B}(2)=N_{1}(m_{2})=2$, $\\#\mathcal{B}(3)=N_{1}(m_{3})=4$,
$\\#\mathcal{B}(4)=N_{1}(m_{4})=6$, and $\\#\mathcal{B}(5)=N_{1}(m_{5})=7$.

According to Formula (5.13) in Corollary 5.1, where $n$ runs from
$n_{\ast}+1=3$ to $n_{\ast}+\ell_{\ast}+\ell+0=6$, the tree width is the
maximum $\mathrm{wd}(\mathcal{T})=7$ of the expressions $N_{1}(m_{2})=2$,
$N_{1}(m_{3})=4$, $N_{1}(m_{4})=6$, and $N_{1}(m_{5})=7$.

The information content of the tree is given by Formula (5.17) in the
Definition 5.3:
$\mathrm{IC}(\mathcal{T})=(\\#\mathcal{B}(2)+\\#\mathcal{B}(3))+(\\#\mathcal{B}(4)+\\#\mathcal{B}(5))=(2+4)+(6+7)=19$.

The algebraic invariants in Table 11, that is, defect of commutativity $k$,
depth $\mathrm{dp}$, derived length $\mathrm{dl}$, abelian type invariants of
the centre $\zeta$, relation rank $\mu$, nuclear rank $\nu$, abelian quotient
invariants $\tau(1)$ of the first maximal subgroup, respectively $\tau_{2}$ of
the commutator subgroup, transfer kernel type $\varkappa$, action flag
$\sigma$, and the factorized order $\\#\mathrm{Aut}$ of the automorphism group
have been computed by means of program scripts written for MAGMA [17].

Each group is characterized by the parameters of the normalized representative
$G_{a}^{n}(z,w)$ of its isomorphism class, according to Formula (9.1), and by
its identifier $\langle 3^{n},i\rangle$ in the SmallGroups Database [19].

The column with header $\\#$ contains the number of groups with identical
invariants (except the presentation and identifier), for each row. ∎

######  Corollary 13.1.

(Actions and relation ranks.)
The algebraic invariants of the vertices of the structured coclass-$1$ tree
$\mathcal{T}^{1}{R}$ with abelian root $R=\langle 9,2\rangle\simeq C_{3}\times
C_{3}$, which is drawn in Figure 17, are listed in Table 11. In particular:

  1. (1)

The groups with $V_{4}$-action are the root $R$, all mainline vertices
$G_{0}^{n}(0,0)$, $n\geq 3$, and the terminal vertices $G_{0}^{n}(\pm 1,0)$
with even logarithmic order $n\geq 4$.

  2. (2)

With respect to the transfer kernel types, all mainline groups of type
$\mathrm{a}.1^{\ast}$, $\varkappa=(0000)$, and the leaves of type
$\mathrm{a}.3$, $\varkappa\sim(2000)$, with odd class $c=n-1\geq 3$, possess a
$V_{4}$-action.

  3. (3)

The relation rank is given by $\mu=4$ for the mainline vertices
$G_{0}^{n}(0,0)$, $n\geq 3$, $\mu=2$ for the terminal extraspecial group
$G_{0}^{3}(0,1)$, and $\mu=3$ otherwise.

  4. (4)

All terminal vertices with odd class, and the mainline vertices with even
class, possess an RI-action. The terminal vertices with odd class are
Schur$+1$ $\sigma$-groups [29, 30].

###### Proof.

(of Corollary 13.1) The existence of an RI-action on $G$ has been checked by
means of an algorithm involving the $p$-covering group of $G$, written for
MAGMA [17]. The other claims follow immediately from Table 11, continued
indefinitely with the aid of the periodicity in Proposition 13.1. A Schur$+1$
$\sigma$-group has an RI-action and relation rank $\mu\leq 3$ [29, 30]. ∎

##  14\. Conclusion

In the core sections 11 and 12 of this paper, we have elaborated our long
desired proof that the pruned tree of all finite $3$-groups with elementary
bicyclic commutator quotient, which do not arise as descendants of non-
metabelian groups, can be described with a finite amount of data.

######  Theorem 14.1.

(Main Theorem on the Finite Information Content.)
The total information content of the coclass forest $\mathcal{F}(r)$ is given
by

(14.1) $\mathrm{IC}(\mathcal{F}(r))=\begin{cases}19&\text{ for }r=1,\\\
739=515+224&\text{ for }r=4,\\\ 501=207+294&\text{ for each odd }r\geq 5,\\\
581=357+224&\text{ for each even }r\geq 6.\end{cases}$

###### Proof.

The total information content of a coclass forest is the sum of the
cardinality of its sporadic part $\mathcal{F}_{0}(r)$ and the information
contents of its pairwise non-isomorphic coclass trees
$\mathcal{T}^{r}R_{i}^{r}$

$\mathrm{IC}(\mathcal{F}(r))=\\#\mathcal{F}_{0}(r)+\sum_{i}\,\mathrm{IC}(\mathcal{T}^{r}R_{i}^{r}).\qed$

Due to the exceptional complexity of the pre-periodic forests $\mathcal{F}(2)$
and $\mathcal{F}(3)$, their information contents are unknown up to now.
However, since they are certainly finite, this does not obfuscate our clear
and beautiful results concerning the infinitely many co-periodic forests
$\mathcal{F}(r)$ with $r\geq 4$, which can be reduced to the finite
information content of the primitive co-period
$(\mathcal{F}(5),\mathcal{F}(6))$.

##  15\. Acknowledgement

We gratefully acknowledge that our research was supported by the Austrian
Science Fund (FWF): Project P 26008-N25. Indebtedness is expressed to the
anonymous referees for valuable suggestions concerning the readability and, in
particular, for drawing our attention to the paper [6].

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