Denote by the rooted tree of all finite 3-groups G with elementary bicyclic commutator quotient, and let be the infinite pruned subtree of, where all descendants of capable non-metabelian vertices are eliminated. The main intention of this paper is to prove that the information content of the tree can be reduced to a finite set of representatives with the aid of two kinds of periodicity.

• Firstly, the well-known virtual periodicity isomorphisms between the finite depth-pruned branches, , of a coclass subtree are refined to strict periodicity isomorphisms between complete branches which reduce the information content of the infinite coclass subtree to the finite union of pre-period and first primitive period. The virtual periodicity was proved by du Sautoy [1] and independently by Eick and Leedham-Green [2] for groups of any prime power order. The strict periodicity for and type is proved in the present paper.

• Secondly, evidence is provided of co-periodicity isomorphisms between the infinite coclass forests, , which reduce the information content of the pruned tree to the union of pre-period and first primitive period, consisting of the leading six coclass forests only. The discovery of this co-periodicity is the progressive innovation in the present paper.

Together with the coclass theorems of Leedham-Green [3] and Shalev [4] , which imply that each coclass forest consists of a finite sporadic part and a finite number of coclass trees, , each having a finite information content due to the strict periodicity, this shows that the pruned infinite subtree of the tree is described by finitely many representatives only.

We begin with a general theory of descendant trees of finite p-groups with arbitrary prime p in §2 and we explain the conceptual foundations of the virtual periodicity isomorphisms between the finite branches of coclass subtrees [1] [2] [5] and the recently discovered co-periodicity isomorphisms between infinite coclass forests in §3. The behavior of algebraic invariants of the tree vertices and their automorphism groups is described with simple transformation laws in §4. The graph theoretic preliminaries are supplemented by connections between depth, width, information content and numbers of immediate descendants in §5, identifiers of groups in §6, and precise definitions of mainlines and sporadic parts in §7. The main theorems are presented in §8.

Then we focus on the tree of finite 3-groups G with abelianization. The flow of our investigations is guided by §10 concerning the remarkable infinite main trunk of certain metabelian vertices in which gives rise to the top vertices of all coclass forests, , by periodic bifurcations and constitutes the germ of the newly discovered co-periodicity of length two. To start with a beautiful highlight, we immediately celebrate the simple structure of the first primitive period in §§11 and 12 and defer the somewhat arduous task of describing the exceptional pre-period to the concluding §§13 and 14.

Finally, we point out that our theory, together with the investigations of Eick [6] , provides an independent verification and confirmation of all results about the metabelian skeleton of the tree in the dissertation of Nebelung [7] , since is a subtree of the pruned tree. The present paper shows the co-periodicity of the sporadic parts and coclass trees, , of the coclass forests, and ( [6] , §5.2, pp. 114-116) establishes the connection between the coclass trees and infinite metabelian pro-3 groups of coclass r.

Let p be a prime number. In the mathematical theory of finite groups of order a power of p, so-called p-groups, the introduction of the parent-child relation by Leedham-Green and Newman ( [8] , pp. 194-195) has simplified the classification of such groups considerably. The relation is defined in terms of the lower central series of a p-group G, where

in particular, is the commutator subgroup of G. Since the series becomes stationary,

a non-trivial p-group is nilpotent of class.

Definition 2.1. If G is non-abelian, then the class- quotient

is called the parent of G, and G is a child (or immediate descendant) of.

Parent and child share a common class-1 quotient (or derived quotient or abelianization), since

according to the isomorphism theorem. The lower central series of is shorter by one term:

and thus.

Definition 2.2. For an assigned finite p-group, the descendant tree with root R is defined as the digraph whose set of vertices V consists of all isomorphism classes of p-groups G with, for some, and whose set of directed edges E consists of all child-parent pairs

The mapping is called the parent operator.

If the root R is abelian, then all vertices of the tree share the common abelianization. Since a nilpotent group with cyclic abelianization is abelian, the descendant tree of a cyclic root consists of the single isolated vertex R. The classification of p-groups by their abelianization is refined further, if directed edges are restricted to starting vertices G with cyclic last non-trivial lower central of order p. Then the descendant tree of R splits into a countably infinite disjoint union

of directed subgraphs, where and the vertices of the component with fixed share the same coclass, , as a common invariant, since the logarithmic order and the nilpotency class of the parent and child G satisfy the rule

Definition 2.3. Thus, the components with are called the coclass subgraphs of the descendant tree.

According to the coclass theorems by Leedham-Green [3] and Shalev [4] , a coclass graph is the disjoint union of a finite sporadic part and finitely many coclass trees (with infinite mainlines), that is, a forest for which there exist integers such that

Definition 2.4. In the present paper, the focus will lie on finite p-groups with fixed prime arising as descendants of the fixed elementary bicyclic 3-group of order with, where denotes the cyclic group of order n. This assumption permits a simplified notation by omitting the explicit mention of p and R. Further, we shall slightly reduce the complexity of the forests, , by eliminating the descendants of capable (i.e., non-terminal) non-metabelian vertices. This pruned light-weight version of will be denoted by, called the coclass-r forest, and Formula (2.9) becomes

and possibly different integers and.

Remark 2.1. In §§11 and 12 it will turn out that the coclass trees with metabelian mainlines do not contain any capable non-metabelian vertices. So the pruning process from to concerns the sporadic part, and reduces the number of coclass trees by eliminating those with non-metabelian mainlines entirely, but does not affect the coclass trees with metabelian mainlines, which remain complete in spite of pruning.

In general, we denote a graph as a pair with set of vertices V and set of edges E.

Definition 3.1. Let and be two digraphs with directed edges in, respectively. If there exists a bijection such that

then and are called isomorphic digraphs, and is an isomorphism of digraphs.

When is a finite digraph with vertex cardinality, we can identify V with the set. Then the set of directed edges is characterized uniquely by the characteristic function of E in, which is called the adjacency matrix of. Its entries are defined, for all, by

Proposition 3.1. Let and be two finite digraphs with n vertices. Then and are isomorphic if and only if there exists a bijection such that the entries of the adjacency matrices coincide for all.

Proof. The bijection satisfies the condition in Formula (3.1) if and only if ↔↔↔.

The in-resp. out-degree of a vertex in a finite digraph can be expressed in terms of the vth column-resp. row-sum of the adjacency matrix:

In particular, if is a finite directed in-tree with root R, then each row of the adjacency matrix A corresponding to a vertex contains a unique 1 and

Proposition 3.2. Let and be two rooted directed in-trees, and denote by and their parent operators. Then a bijection with is an isomorphism of rooted directed in-trees if and only if for all, that is, (briefly: commutes with the parent operator), as shown in Figure 1.

Proof. Recall that each row of the adjacency matrix A of the tree corresponding to a vertex, , contains a unique 1. This fact can be used to define the parent operator of by Û. Consequently, if has the claimed property to commute with the parent operator, then Û Û Û Û Û Û. For infinite trees, the steps concerning adjacency matrices must be omitted. The proof of the converse statement is similar.

The graph theoretic structure of a coclass tree with unique infinite mainline and finite branches, consisting of isomorphism classes of finite p-groups, is described by the following concepts.

Definition 5.1. Let be a coclass tree. Suppose that the tree root R is of logarithmic order, and denote by the unique mainline vertex with. In particular,.

For, the difference set is called the eth branch of.

Let be one of the branches of. For any integer, we let

denote the nth layer of, respectively.

The width of the tree is the maximal cardinality of its layers,

Each vertex v of the branch is connected with the mainline by a unique finite path of directed edges from v to the branch root, formed by the iterated parents of v,

The length of this path is called the depth of v.

The depth of a branch is the maximal depth of its vertices,

Definition 5.2. Let be a coclass tree. The depth of the tree is the maximal depth of its branches,

Throughout this paper, we assume that both, the depth and the width of the tree, are bounded. This assumption is satisfied by all trees of finite 3-groups under investigation in the sequel. However, we point out that that tree of finite 5-groups with coclass one has unbounded depth, and the tree of finite 7-groups with coclass one even has unbounded width and depth. (Compare [6] , 5.1, pp. 113-114)

Lemma 5.1. Let, and. Then

Proof. Since is the root of the branch, we have, but . Since, there exists a vertex, necessarily terminal if, such that. The iterated parents of t form the unique finite path from t to the branch root (see Figure 3),

and we have but .

Lemma 5.2. Let and. Then

Proof. Since, we have for each. A branch with cannot contribute to. On the other hand, if, then a branch with cannot contribute to either, since, according to Lemma 5.1, and we obtain (see Figure 3). Consequently,

Since the implementation of the p-group generation algorithm [12] [13] [14] in the computational algebra system MAGMA [15] [16] [17] is able to give the number of all, respectively only the capable, immediate descendants (children) of an assigned finite p-group, we express the cardinalities of the branches of a coclass tree, which were given in a preliminary form in Lemma 5.1, in terms of these numbers, respectively.

Theorem 5.1. Let be a coclass tree with tree root R of logarithmic order, pre-period of length, and period of primitive length. For each vertex, denote by the number of all children (of step size) and by the number of capable children of v. When is the vertex with on the mainline of, let with be the capable children of, in particular, let be the next mainline vertex. Finally, let with denote the capable children of, for each.

1) If the tree is of depth, then

2) If the tree is of depth, then

3) If the tree is of depth, then

Proof. Put. Generally, we have with, according to Lemma 5.1.

If, then and. We have and, since the next mainline vertex is one of the children of but does not belong to. Thus, we obtain .

If, then and, where and as before, and. Therefore,.

If, then and, where, , as before, and . Thus,

Remark 5.1. In Theorem 5.1, item (1) is included in item (2), since implies, and item (2) is included in item (3), since implies, for all.

Corollary 5.1. Under the same assumptions as in Theorem 5.1, the width of the coclass tree, in dependence on the depth and the periodicity, is generally given by

For assigned small values of the depth, the width can be expressed in terms of descendant numbers in the following manner:

1) If the tree is of depth, then

2) If the tree is of depth, then is the maximum among the number and all expressions

where n runs from to.

3) If the tree is of depth, then is the maximum among the numbers, , and all expressions

where n runs from to.

Proof. According to [1] [2] [5] , the periodicity of the branches of a coclass tree with root and bounded depth and width can be expressed by means of isomorphisms between branches, starting from the periodic root:

where denotes the length of the pre-period and is the primitive period length. With Lemma 5.1, an immediate consequence is the periodicity of branch layer cardinalities:

According to Lemma 5.2, we have, and thus

For finding the maximal layer cardinality, the root term can be omitted, since each layer contains a mainline vertex. Beginning with, the expression for the tree layer cardinality is a sum of terms and we must find the logarithmic order where periodicity of all terms sets in. This leads to the inequality with solution. Consequently, is the biggest logarithmic order for which a new value of the tree layer cardinality may occur (see Figure 3). At the logarithmic order, periodic repetitions of the values of tree layer cardinalities begin.

In the special case of, Theorem 5.1 yields an expression in terms of descendant numbers:

The following concept provides a quantitative measure for the finite infor- mation content of an infinite tree with periodic branches.

Definition 5.3. By the information content of a coclass tree we understand the sum of the cardinalities of all branches belonging to the pre-period and to the primitive period of,

where denotes the logarithmic order of the periodic root P of (see Figure 3).

Independently of being metabelian or non-metabelian, a finite 3-group G of order up to will be characterized by its absolute identifier , according to the SmallGroups Database [18] [19] . Starting with order, a group G is characterized by the absolute identifier of the parent in the SmallGroups Database [19] together with a relative identifier generated by the ANUPQ package [20] of MAGMA [17] . Here, s denotes the step size of the directed edge. Occasionally, certain groups of order and coclass 2 are identified by single capital letters similarly as in [21] [22] [23] .

If we define a mainline as a maximal path of infinitely many directed edges of step size, then there arises the ambiguity that a vertex could be root of several coclass trees. The metabelian 3-group, for instance, would be the end vertex of more then one mainline, namely on the one hand of the metabelian mainline

and on the other hand of non-metabelian mainlines, one which ends with

and three which end with,.

Therefore, an additional condition is required in the precise definition of a mainline.

Definition 7.1. A mainline is a maximal path of infinitely many equally oriented edges of step size, in none of whose vertices other infinite paths of step size are ending.

The end vertex of a mainline is called the root of a coclass tree.

Definition 7.1 can be expressed equivalently in terms of infinite pro-p groups ( [6] , 3.1, p. 107).

Example 7.1. The metabelian 3-group is root of the coclass-2 tree with metabelian mainline.

The metabelian 3-group is root of the coclass-2 tree with non-metabelian mainline. According to our pruning convention that descendants of capable non-metabelian vertices do not belong to the coclass forests, this tree is not an object of examination in the present paper.

Finally, the metabelian 3-group is not root of a coclass tree.

Based on the precise definition of a mainline and a root of a coclass tree, we are now in the position to give an exact specification of the sporadic part of a coclass forest.

Definition 7.2. The sporadic part of the coclass forest with is the complement of the union of the (finitely many) coclass trees in the forest,

There is no necessity, to restrict the concepts of a mainline, a coclass tree and its root further by stipulating the coclass stability of the root. It is therefore admissible that directed edges of step size end in vertices (mainline or of depth) of a coclass tree, due to the phenomenon of multifurcation.

Example 7.2. In the second mainline vertex of the coclass-2 tree with root, a bifurcation occurs, due to the nuclear rank. In fact, the directed edge of step size which ends in the vertex is the final edge of an infinite path with alterating step sizes and, due to periodic bifurcations. However, this non-meta- belian path is not the topic of investigations in the present paper. For detailed information on these matters see [24] and [25] .

The same is true for the second mainline vertex of the coclass-2 tree with root.

The unnecessary requirement of coclass stability would eliminate the pre-periods of the trees and and enforce purely periodic subtrees with periodic coclass-settled roots, namely and .

An important technique in the theory of descendant trees is to reduce the structure of an infinite tree to a periodically repeating finite pattern. In particular, it is well known [1] [2] [5] that an infinite coclass tree of finite p-groups with fixed coclass is the disjoint union of its branches

, which can be partitioned into a single finite pre-period

of length and infinitely many copies of a finite primitive period of length, where the integer characterizes the position of the periodic root on the mainline, provided the tree is suitably depth-pruned.

The following first main result of this paper establishes the details of the primitive period of branches of five coclass-4 trees, with, respectively of three coclass-5 trees, with, of finite 3-groups with mainline vertices having a single total transfer kernel and roots with, respec- tively with, writ- ten in the notation of [18] [19] [20] . In fact, we prove more than the virtual periodicity for arbitrary finite p-groups in [1] [2] [5] , since all trees of the particular finite 3-groups in our investigation have bounded depth and therefore reveal strict periodicity.

Theorem 8.1. (Main Theorem on Strict Periodicity Isomorphisms of Branches.)

For each integer, there exists a bijective mapping which is a strict isomorphism of finite structured in-trees for the strict invariants in-degree, out-degree, coclass, relation rank, nuclear rank, action flag, and transfer kernel type. Moreover, is a f-isomorphism of finite structured in-trees for the following -invariants with their transformation laws:

• logarithmic order with,

• nilpotency class with,

• order of the automorphism group with,

• first component of the transfer target type with , and

• commutator subgroup with , respectively .

Consequently, the branches of each tree , respectively , are purely periodic with primitive length at most.

Proof. The -isomorphisms between the finite branches of a tree describe the first periodicity and reduce an infinite tree to its finite primitive period, provided the periodicity is pure. This will be proved for even coclass in Theorem 11.3 for, in Thm. 11.4 for, in Thm. 11.5 for, in Thm. 11.6 for, and in Thm. 11.7 for. For odd coclass, it will be proved in Theorem 12.3 for, in Thm. 12.4 for, and in Thm. 12.5 for.

Invariants connected with the nilpotency class are not strict and satisfy the following transformation laws: the shift for and, and the corresponding transformations for,and for, with fixed coclass r. For, the transformation law is described by the homothety.

Theorems 11.3, 11.4, 11.5, 11.6, 11.7 and 12.3, 12.4, 12.5 will give detailed descriptions of the structure of these trees, in particular they will establish a quantitative measure for the finite information content of each tree.

Remark 8.1. According to Theorem 8.1, the diagrams of coclass-r trees whose mainline vertices V possess a single total kernel among the transfers to the four maximal subgroups reveal several surprising features: firstly, the branches are purely periodic of primitive length at most 2 without pre-period, secondly, the branches are of uniform depth 2 only, and finally, none of the vertices gives rise to descendants of coclass bigger than r. So the trees are entirely regular and coclass-stable, in contrast to the trees with 3-groups G of coclass as vertices.

Unfortunately it is much less well known that the entire metabelian skeleton of the descendant tree of the elementary bicyclic 3-group is the disjoint union of its coclass subgraphs , where each component consists of a finite sporadic part and finitely many metabelian coclass trees, and there is a periodicity for each. This was proved by Nebelung [7] and confirmed by Eick ( [6] , Cnj. 14, p. 115).

The following second main result of this paper extends the periodicity from the metabelian skeleton to the entire descendant tree, including all the non-metabelian vertices, provided the mainline vertices are still metabelian. Here, we include coclass trees of finite 3-groups with mainline vertices having two total transfer kernels and roots with, respectively with.

Theorem 8.2. (Main Theorem on Co-Periodicity Isomorphisms of Coclass Trees.)

Let the integer be an upper bound. For each integer, and for each of the six roots, , with even coclass, respectively the four roots, , with odd coclass, there exists a bijective mapping, respectively, which is a strict isomorphism of infinite structured in-trees for the strict invariants in-degree, out-degree, relation rank, nuclear rank, action flag, and transfer kernel type. Moreover, is a f-isomorphism of infinite structured in-trees for the following f-invariants with their transformation laws f:

• logarithmic order with,

• nilpotency class with,

• coclass with,

• order of the automorphism group with,

• first component of the transfer target type with , and

• commutator subgroup with , respectively .

Proof. The statement for the metabelian skeletons of the coclass trees is one of the main results of Nebelung’s thesis [7] . With the aid of Theorem 8.1, the periodicity of the entire coclass trees with and fixed subscript i has been verified by computing the metabelian and non-metabelian vertices of the first four branches with of the trees. The computations were executed by running our own program scripts for the Computer Algebra System MAGMA [17] , which contains an implementation of the p-group generation algorithm by Newman [12] [26] and O’Brien [13] [14] , the SmallGroups Database [18] [19] , and the ANUPQ package [20] . It turned out that, firstly, and, for each, and secondly, and, for each.

The established -isomorphisms between the infinite coclass trees and, for, describe the germ of the second periodicity expressed in Conjecture 8.1. Invariants connected with the nilpotency class or coclass are not strict and are subject to the following mappings: the shifts for,for,and for,and the corresponding transformationsfor,andfor.For,the transformation law is described by the homothety.

Thus, the confidence in the validity of the following conjecture is supported extensively by sound numerical data.

Conjecture 8.1. (Co-Periodicity Isomorphisms of All Coclass-r Trees for.)

Theorem 8.2 remains true when the upper bound is replaced by any upper bound.

Consequently, all coclass trees with and fixed subscript i are co-periodic in the variable coclass parameter r with primitive length. The eight coclass trees with, and for, for, for, can be viewed as the pre-period of the co-periodicity. (Compare [6] , Cnj. 14, p. 115).

The general graph theoretic and algebraic foundations of the coclass forests with have been developed completely in the preceding Sections 2 - 7. Now we can turn to the main goal of the present paper, that is, the proof of the main theorems in section 8 by the systematic investigation of finite 3-groups G with commutator quotient, represented by vertices of the descendant tree, with the single restriction that the parent of G is metabelian. To this end, we first need parametrized presentations for all metabelian vertices of.

The flow of our investigations is guided by the present section concerning the remarkable infinite main trunk of certain metabelian vertices in which gives rise to the top vertices of all coclass forests by periodic bifurcations and constitutes the germ of the newly discovered co-periodicity of length two. Since the minimal possible values of the nilpotency class and logarithmic order of a finite metabelian 3-group with coclass, belonging to the forest, are given by and, it follows that G must be an immediate descendant of step size of its parent. The crucial fact is that this parent is precisely the vertex with of the main trunk. In the following, we rather use the coclass j of the parent than r of the children.

Theorem 10.1. (The main trunk.)

1) In the descendant tree of the abelian root, there exists a unique infinite path of (reverse) directed edges such that, for each fixed coclass, every metabelian 3-group G with and is a proper descendant of.

2) The trailing vertex is exactly the extra special Blackburn group with exceptional transfer kernel type (TKT ) a.1,.

3) All the other vertices with share the common TKT b.10, , possess nilpotency class, coclass, logarithmic order, abelian commutator subgroup of type, and transfer target type, where.

4) For, periodicity of length 2 sets in, has nuclear rank, relation rank, and immediate descendant numbers (including non-metabelian groups)

Restricted to metabelian groups, the immediate descendant numbers are

All immediate descendants are -groups, if is odd, but only, if, and, if is even.

Proof. See the dissertation of Nebelung ( [7] , p. 192).

Remark 10.1. Although the number of metabelian children of step sizes of the vertex with fit into the periodic pattern , the number of all children of step sizes of is bigger than usual with instead of. Therefore, periodicity starts with and not with.

Corollary 10.1. (All coclass trees with metabelian mainlines.)

The coclass trees of 3-groups G with, whose mainlines consist of metabelian vertices, possess the following remarkable periodicity of length 2, drawn impressively in Figure 4.

1) For even, the vertex with subscript of the main trunk has exactly 4 immediate descendants of step size giving rise to coclass trees whose mainline vertices are metabelian 3-groups G with odd and fixed TKT, either d.19, , or d.23, , or d.25, , or b.10, , the latter with root.

2) For odd, the vertex with subscript of the main trunk has exactly 6 immediate descendants of step size giving rise to coclass trees whose mainline vertices are metabelian 3-groups G with even and fixed TKT, either d.19, , twice, or d.23, , or d.25, , twice, or b.10, , the latter with root.

3) The unique pre-periodic exception is the vertex of the main trunk, which has exactly 3 immediate descendants of step size giving rise to coclass trees whose mainline vertices are metabelian 3-groups G with even and fixed TKT, either c.18, , or c.21, , or b.10, , the latter with root.

Proof. See the dissertation of Nebelung ( [7] , 5.2, pp. 181-195).

Although formulated for the particular coclass, all results for periodic groups and most of the results for sporadic groups in this section are valid for any even coclass. The only exception is the bigger (and thus pre-periodic) sporadic part of the coclass forest, described in Proposition 11.2, whereas the (co-periodic) standard case, the sporadic part of the coclass forest, is presented in Proposition 11.1.

Figure 5 sketches an outline of the metabelian skeleton of the coclass forest in its top region. The vertices and,

with the crucial bifucation from to, belong to the infinite main trunk (§10).

Proposition 11.1 (Co-periodic standard case.)

The sporadic part of the coclass-6 forest consists of

• 13 isolated metabelian vertices of order 3¹³ with types F.7, F.11, F.12, F.13,

• 8 metabelian roots of finite trees with types G.16, G.19, H.4, together with a metabelian child having a GI-action, which is unique for each root, and 22 metabelian and 38 non-metabelian children without GI-action, all with depth and,

• 66 isolated vertices with and types d.19, d.23, d.25,

• 179 isolated vertices with and type b.10,

• 23 capable vertices with and type b.10,

Whose children do not belong to, by definition.

The action flag of all metabelian top vertices with depth is. The value only occurs for all vertices with type b.10, d.25, G.19, and certain vertices with type G.16, H.4, but never for type d.19, d.23, F.7, F.11, F.12, F.13. Exactly the isolated vertices with depth have an RI-action.

Together with the 6 metabelian roots, , of coclass-6 trees, the top vertices of depth of are exactly the children of step size of the main trunk vertex, and the capable vertices among them correspond to the invariant of.

Proposition 11.2. (Pre-periodic exception.)

The constitution of the sporadic part of the coclass-4 forest with respect to the 21 metabelian top vertices and their 68 children (here with order 3⁹, resp.) is the same as described for in Proposition 11.1, but the number of non-metabelian top vertices of depth is bigger, namely

• 88 isolated vertices with and types d.19, d.23, d.25,

• 12 capable vertices with and types d.19, d.23, d.25,

whose children do not belong to, by definition,

• 268 isolated vertices with and type b.10,

• 58 capable vertices with and type b.10,

whose children do not belong to, by definition.

The distribution of the action flags is the same as in Proposition 11.1, but the total census of top vertices is considerably bigger:

Together with the 6 metabelian roots, , of coclass-4 trees, the top vertices of depth of are exactly the children of step size of the main trunk vertex, and the capable vertices among them corres- pond to the invariant of.

Theorem 11.1. The coclass-r forest with any even is the disjoint union of its finite sporadic part with total information content

and infinite coclass-r trees with roots, where

The algebraic invariants for groups with positive action flag, and in cumulative form for, are given for in Table 1, where the parent vertex on the main trunk is also included, but the 426 non-metabelian top vertices of depth are excluded.

Proof. (of Propositions 11.1, 11.2, and Theorem 11.1) We have computed the sporadic parts of coclass forests with even up to by means of MAGMA [17] . Except for the differences pointed out in the