Research Frontier 2013

Descendant Trees of Finite p-Groups

a phenomenon connected with

Nuclear Ranks, Schur Multipliers and Cohomology



Karl-Franzens University Graz, left side

Introduction:

  • Here, our aim is firstly
    to give conditions for a finite p-group
    to be a terminal leaf or
    to give rise to descendants
    in a graph or coclass tree.

    Secondly, due to its fundamental
    importance in algebraic number theory,
    we want to address the problem
    whether a finite p-group G is admissible
    for being the Galois group
    G = Gal(Fp(K)|K) of the maximal
    unramified pro-p extension of various
    types of algebraic number fields K.

  • The tree structure arises by giving the
    uniquely determined parent π(G) of G,
    connected by an edge of depth 1 with G,
    or the generalized parent (of depth at least 2),
    indicated by red font.

  • Starting with some basic invariants,
    we denote by pn the order of G,
    by cl the class of G,
    by cc the coclass of G,
    such that n = cl + cc,
    by Z(G) the centre of G, and
    by dl the derived length of G.
    The group G itself is identified by its
    SmallGroups Library ID
    of GAP and MAGMA .

  • Next we give two advanced invariants
    which are able to identify
    a finite batch of candidates,
    most frequently a unique candidate,
    for the metabelianization
    G = Gal(Fp2(K)|K) = H/H''
    of the p-tower group
    H = Gal(Fp(K)|K):
    the transfer kernel type (TKT)
    κ(G) = (Ker(Ti))1 ≤ i ≤ p+1
    and the transfer target type (TTT)
    τ(G) = (Mi/Mi')1 ≤ i ≤ p+1
    of the Artin transfers Ti:G/G'→Mi/Mi'
    with respect to the maximal normal
    subgroups Mi of G.

  • Aut denotes the factorized order of the
    automorphism group of G and
    σ = 1 indicates the existence of
    an automorphism of G
    which acts as inversion on G/G',
    whence Aut is divisible by 2.
    For a quadratic field K and an odd prime p,
    the condition σ = 1 to be a σ-group
    is necessary for G = Gal(Fpm(K)|K),
    for any m ≥ 2 and for m = ∞.

  • Finally, the column Rank shows
    the nuclear rank N of G and
    the p-multiplier rank M of G,
    given in the format N/M.
    G is terminal iff N = 0,
    indicated by green font.
    N = 0 implies that
    the Schur multiplier of G is trivial.
    For N = 1, there exist descendants
    of the same coclass.
    For N ≥ 2, however, G is not coclass settled,
    indicated by red font.
    Column Desc lists the numbers of descendants
    with increasing coclass in the format
    "all descendants / capable descendants".
    Cohomology comes in via the
    generator rank d = dimp H1(G,Fp)
    and relation rank r = dimp H2(G,Fp).
    G is a Schur group iff r = d and a
    Schur+n group iff d ≤ r ≤ d+n,
    for some n ≥ 0.
    A Schur group is also called
    a group with balanced presentation,
    and we have r = d iff N = 0, M = d.


Karl-Franzens University Graz, centre with 8 figures

Basically, we intend to find non-metabelian Schur+n σ-groups for 0 ≤ n ≤ 1
by starting from one of the abelian roots A in {(3,3),(3,9),(5,5)} and using the ANUPQ package
of MAGMA to construct the relevant parts of their descendant trees T(A).
We mainly, but not exclusively, restrict our investigations to σ-groups having σ = 1.
3-groups are considered in 1 and 5-groups in 2.


1. Coclass Trees of 3-Groups G:


We begin with our (meanwhile well-known) joint discoveries with M. R. Bush in 1.1,
and we generalize the ideas to non-elementary abelianizations in 1.2, keeping the prime p = 3 fixed,
and to the next prime p = 5 in 2.1, keeping the elementary abelianization fixed.


1.1. Abelianization G/G' of Type (3,3):


Root of coclass graph G(3,1) is the 3-elementary bicyclic group
<32,2> ≅ (3,3), which is not coclass settled.
Exponent-3 descendant is the abelian group <33,2> ≅ (3,9).
Generalized descendants of depth 2 are
the abelian group <34,2> ≅ (9,9)
and the groups <34,3|4> with abelianization (3,9).
Generalized descendant of depth 3 is the group <35,2> with abelianization (9,9).


It is well known that the unique tree of coclass graph G(3,1) has branches of depth 1
and entirely consists of metabelian groups G with G/G' ≅ (3,3).
We show that the smallest Schur+1 σ-groups occur on branch B(3) of this tree
but then we leave G(3,1) and proceed to groups G with cc(G) ≥ 2 and G/G' ≅ (3,3).


There are only 2 metabelian Schur σ-groups, <35,5> and <35,7>, which are of coclass 2.
Schur σ-groups G of derived length dl(G) = 3 start with order |G| = 38 and coclass cc(G) = 3,
forming Schur covers G of unbalanced σ-groups H ≅ G/G''.


G π(G) cl cc Z(G) TKT Mi TTT G'/G'' dl Aut σ Rank Desc
|G| = 9
<32,2> <1,1> 1 1 (3,3) a.1 (0000) <31,1>4 (3)4 1 1 2431 1 3/3 3/2;3/3;1/1
|G| = 27
<33,3> <32,2> 2 1 (3) a.1 (0000) <32,2>4 (3,3)4 (3) 2 2433 1 2/4 4/1;7/5
<33,4> <32,2> 2 1 (3) A.1 (1111) <32,2>,<32,1>3 (3,3),(9)3 (3) 2 2133 0 0/2 0/0
|G| = 81
<34,7> <33,3> 3 1 (3) a.3* (2000) <33,5>,<33,3>,<33,4>2 (3,3,3),(3,3)3 (3,3) 2 2234 1 0/3 0/0
<34,8> <33,3> 3 1 (3) a.3 (2000) <33,2>,<33,3>,<33,4>2 (3,9),(3,3)3 (3,3) 2 2234 1 0/3 0/0
<34,9> <33,3> 3 1 (3) a.1 (0000) <33,2>,<33,3>3 (3,9),(3,3)3 (3,3) 2 2235 1 1/4 6/1
<34,10> <33,3> 3 1 (3) a.2 (1000) <33,2>,<33,4>3 (3,9),(3,3)3 (3,3) 2 2135 1 0/3 0/0
|G| = 243
<35,3> <33,3> 3 2 (3,3) b.10 (2100) <34,12>2,<34,3>2 (3,3,3)2,(3,9)2 (3,3,3) 2 2336 1 2/4 10/6;15/15
<35,4> <33,3> 3 2 (3,3) H.4 (4111) <34,12>,<34,13>2,<34,3> (3,3,3)3,(3,9) (3,3,3) 2 2236 1 1/3 4/4
<35,5> <33,3> 3 2 (3,3) D.10 (3144) <34,3>,<34,13>,<34,3>2 (3,9),(3,3,3),(3,9)2 (3,3,3) 2 2136 1 0/2 0/0
<35,6> <33,3> 3 2 (3,3) c.18 (0122) <34,3>,<34,12>,<34,3>2 (3,9),(3,3,3),(3,9)2 (3,3,3) 2 2236 1 1/3 4/4
<35,7> <33,3> 3 2 (3,3) D.5 (1133) <34,3>,<34,13>,<34,3>,<34,13> (3,9),(3,3,3),(3,9),(3,3,3) (3,3,3) 2 2236 1 0/2 0/0
<35,8> <33,3> 3 2 (3,3) c.21 (2034) <34,3>4 (3,9)4 (3,3,3) 2 2236 1 1/3 4/4
<35,9> <33,3> 3 2 (3,3) G.19 (2143) <34,3>4 (3,9)4 (3,3,3) 2 2436 1 1/3 2/2
|G| = 729
<36,34> = H <35,3> 4 2 (3) b.10 (2100) <35,37>2,<35,13>2 (3,3,3)2,(3,9)2 (3,3,3,3) 2 2338 1 2/5 6/2;15/8
<36,35> = I <35,3> 4 2 (3) b.10 (2100) <35,37>,<35,38>,<35,13>2 (3,3,3)2,(3,9)2 (3,3,3,3) 2 2138 0 1/4 6/6
<36,37> = A <35,3> 4 2 (3) b.10 (2100) <35,53>2,<35,14>2 (3,3,3)2,(3,9)2 (3,3,9) 2 2338 1 2/5 6/0;15/3
<36,38> = C <35,3> 4 2 (3) b.10 (2100) <35,53>,<35,54>,<35,14>2 (3,3,3)2,(3,9)2 (3,3,9) 2 2138 0 1/4 5/0
<36,40> = B <35,3> 4 2 (3,3) b.10 (2100) <35,53>2,<35,14>,<35,2> (3,3,3)2,(3,9),(9,9) (3,3,9) 2 2238 1 2/5 16/2;27/4
<36,#1;2> = <36,45> <35,4> 4 2 (3) H.4 (4111) <35,53>,<35,42>2,<35,14> (3,3,3)3,(3,9) (3,3,3) 2 2238 1 2/4 4/0;2/1
<36,49> = Q <35,6> 4 2 (3,3) c.18 (0122) <35,2>,<35,53>,<35,15>2 (9,9),(3,3,3),(3,9)2 (3,3,9) 2 2238 1 2/4 8/3;6/3
<36,54> = U <35,8> 4 2 (3,3) c.21 (2034) <35,15>,<35,2>,<35,15>2 (3,9),(9,9),(3,9)2 (3,3,9) 2 2238 1 2/4 8/3;6/3
<36,#1;2> = <36,57> <35,9> 4 2 (3) G.19 (2143) <35,13>4 (3,9)4 (3,3,3,3) 2 2438 1 2/4 1/0;6/6
|G| = 2187
<37,260> <36,40> 5 2 (9) b.10 (2100) <36,405>2,<36,18>,<36,24> (3,3,3)2,(3,9),(9,9) (3,3,9) 3 2238 1 0/4 0/0
<37,262> <36,40> 5 2 (9) b.10 (2100) <36,145>,<36,405>,<36,82>,<36,25> (3,3,3)2,(3,9),(9,9) (3,3,9) 3 2138 1 0/4 0/0
<37,#1;1> = <37,270> <36,45> 5 2 (3) H.4 (4111) <36,145>,<36,389>2,<36,73> (3,3,3)3,(3,9) (3,3,9) 3 2238 1 0/3 0/0
<37,#1;2> = <37,271> <36,45> 5 2 (3) H.4 (4111) <36,399>,<36,390>2,<36,17> (3,3,3)3,(3,9) (3,3,9) 3 2239 1 0/3 0/0
<37,#1;3> = <37,272> <36,45> 5 2 (3) H.4 (4111) <36,400>,<36,390>2,<36,17> (3,3,3)3,(3,9) (3,3,9) 3 2139 1 0/3 0/0
<37,#1;4> = <37,273> <36,45> 5 2 (3) H.4 (4111) <36,410>,<36,388>,<36,132>,<36,76> (3,3,3)3,(3,9) (3,3,9) 3 2138 1 0/3 0/0
<37,#1;1> = <37,284> <36,49> 5 2 (9) c.18 (0122) <36,24>,<36,145>,<36,83>2 (9,9),(3,3,3),(3,9)2 (3,3,9) 3 2238 1 0/3 0/0
<37,#1;2> = <37,285> <36,49> 5 2 (3,3) c.18 (0122) <36,23>,<36,395>,<36,79>2 (9,27),(3,3,3),(3,9)2 (3,9,9) 2 22310 1 1/4 5/2
<37,#1;3> = <37,286> <36,49> 5 2 (3,3) H.4 (2122) <36,23>,<36,395>,<36,79>2 (9,27),(3,3,3),(3,9)2 (3,9,9) 2 22310 1 1/4 5/2
<37,#1;4> = <37,287> <36,49> 5 2 (3,3) H.4 (2122) <36,23>,<36,395>,<36,79>2 (9,27),(3,3,3),(3,9)2 (3,9,9) 2 22310 1 1/4 5/2
<37,#1;5> = <37,288> <36,49> 5 2 (3,3) E.6 (1122) <36,23>,<36,396>,<36,79>2 (9,27),(3,3,3),(3,9)2 (3,9,9) 2 21310 1 0/3 0/0
<37,#1;6> = <37,289> <36,49> 5 2 (3,3) E.14 (4122) <36,23>,<36,396>,<36,79>2 (9,27),(3,3,3),(3,9)2 (3,9,9) 2 21310 1 0/3 0/0
<37,#1;7> = <37,290> <36,49> 5 2 (3,3) E.14 (3122) <36,23>,<36,396>,<36,79>2 (9,27),(3,3,3),(3,9)2 (3,9,9) 2 21310 1 0/3 0/0
<37,#1;8> = <37,291> <36,49> 5 2 (9) c.18 (0122) <36,25>,<36,405>,<36,83>,<36,21> (9,9),(3,3,3),(3,9)2 (3,3,9) 3 2138 1 0/3 0/0
<37,#1;1> = <37,301> <36,54> 5 2 (3,3) G.16 (2134) <36,79>,<36,23>,<36,79>2 (3,9),(9,27),(3,9)2 (3,9,9) 2 22310 1 1/4 5/2
<37,#1;2> = <37,302> <36,54> 5 2 (3,3) E.9 (2334) <36,79>,<36,23>,<36,79>2 (3,9),(9,27),(3,9)2 (3,9,9) 2 21310 1 0/3 0/0
<37,#1;3> = <37,303> <36,54> 5 2 (3,3) c.21 (2034) <36,79>,<36,23>,<36,79>2 (3,9),(9,27),(3,9)2 (3,9,9) 2 22310 1 1/4 5/2
<37,#1;4> = <37,304> <36,54> 5 2 (3,3) E.8 (2234) <36,79>,<36,23>,<36,79>2 (3,9),(9,27),(3,9)2 (3,9,9) 2 21310 1 0/3 0/0
<37,#1;5> = <37,305> <36,54> 5 2 (3,3) G.16 (2134) <36,79>,<36,23>,<36,79>2 (3,9),(9,27),(3,9)2 (3,9,9) 2 22310 1 1/4 5/2
<37,#1;6> = <37,306> <36,54> 5 2 (3,3) E.9 (2434) <36,79>,<36,23>,<36,79>2 (3,9),(9,27),(3,9)2 (3,9,9) 2 21310 1 0/3 0/0
<37,#1;7> = <37,307> <36,54> 5 2 (9) c.21 (2034) <36,21>,<36,24>,<36,82>2 (3,9),(9,9),(3,9)2 (3,3,9) 3 2238 1 0/3 0/0
<37,#1;8> = <37,308> <36,54> 5 2 (9) c.21 (2034) <36,83>,<36,25>,<36,82>,<36,18> (3,9),(9,9),(3,9)2 (3,3,9) 3 2138 1 0/3 0/0
<37,#1;1> = <37,311> <36,57> 5 2 (3) G.19 (2143) <36,9>,<36,65>,<36,68>2 (3,9)4 (3,3,3,3) 3 2238 1 0/3 0/0
<37,64> <35,3> 4 3 (3,3) b.10 (2100) <36,292>2,<36,26>2 (3,3,3)2,(9,9),(9,9) (3,3,3,9) 2 23310 1 4/6 33/2;453/84;918/540;198/198
G π(G) cl cc Z(G) TKT Schur cover of TTT G'/G'' dl Aut σ Rank Desc
|G| = 6561
<38,#1;1> <37,285> 6 2 (3,3) c.18 (0122) (27,27),(3,3,3),(3,9)2 (3,9,27) 2 22312 1 1/4 7/3
<38,#1;2> <37,286> 6 2 (3) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 2 22312 1 1/4 3/0
<38,#1;2> <37,287> 6 2 (3) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 2 22312 1 1/4 3/0
<38,#1;4> <37,301> 6 2 (3) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 2 22312 1 1/4 2/0
<38,#1;1> <37,303> 6 2 (3,3) c.21 (2034) (3,9),(27,27),(3,9)2 (3,9,27) 2 22312 1 1/4 7/3
<38,#1;4> <37,305> 6 2 (3) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 2 22312 1 1/4 2/0
<38,#2;1> <36,45> 5 3 (3,3) H.4 (4111) (3,3,3)3,(3,9) (3,3,9) 3 2239 1 1/3 4/4
<38,#2;2> <36,45> 5 3 (3,3) H.4 (4111) a cover of <36,45> (3,3,3)3,(3,9) (3,3,9) 3 2139 1 0/2 0/0
<38,#2;1> <36,49> 5 3 (3,9) c.18 (0122) (9,27),(3,3,3),(3,9)2 (3,9,9) 3 22310 1 1/3 4/4
<38,#2;2> <36,49> 5 3 (3,9) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,9) 3 22310 1 1/3 4/4
<38,#2;3> <36,49> 5 3 (3,9) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,9) 3 22310 1 1/3 4/4
<38,#2;4> <36,49> 5 3 (3,9) E.6 (1122) the unique cover of <37,288> (9,27),(3,3,3),(3,9)2 (3,9,9) 3 21310 1 0/2 0/0
<38,#2;5> <36,49> 5 3 (3,9) E.14 (4122) the unique cover of <37,289> (9,27),(3,3,3),(3,9)2 (3,9,9) 3 21310 1 0/2 0/0
<38,#2;6> <36,49> 5 3 (3,9) E.14 (3122) the unique cover of <37,290> (9,27),(3,3,3),(3,9)2 (3,9,9) 3 21310 1 0/2 0/0
<38,#2;1> <36,54> 5 3 (3,9) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,9) 3 22310 1 1/3 4/4
<38,#2;2> <36,54> 5 3 (3,9) E.9 (2334) the unique cover of <37,302> (3,9),(9,27),(3,9)2 (3,9,9) 3 21310 1 0/2 0/0
<38,#2;3> <36,54> 5 3 (3,9) c.21 (2034) (3,9),(9,27),(3,9)2 (3,9,9) 3 22310 1 1/3 4/4
<38,#2;4> <36,54> 5 3 (3,9) E.8 (2234) the unique cover of <37,304> (3,9),(9,27),(3,9)2 (3,9,9) 3 21310 1 0/2 0/0
<38,#2;5> <36,54> 5 3 (3,9) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,9) 3 22310 1 1/3 4/4
<38,#2;6> <36,54> 5 3 (3,9) E.9 (2434) the unique cover of <37,306> (3,9),(9,27),(3,9)2 (3,9,9) 3 21310 1 0/2 0/0
<38,#2;1 | #2;3 | #2;5> <36,57> 5 3 (3,3) G.19 (2143) (3,9)4 (3,3,3,3) 3 23310 1 1/3 2/2
<38,#2;2 | #2;4> <36,57> 5 3 (3,3) G.19 (2143) (3,9)4 (3,3,3,3) 3 24310 1 1/3 2/2
<38,#2;6> <36,57> 5 3 (3,3) G.19 (2143) (3,9)4 (3,3,3,3) 3 24310 1 2/4 6/6;8/8
|G| = 19683
<39,#1;1> <37,285>-#1;1 7 2 (3,3) c.18 (0122) (27,81),(3,3,3),(3,9)2 (3,27,27) 2 22314 1 1/4 5/2
<39,#1;2> <37,285>-#1;1 7 2 (3,3) H.4 (2122) (27,81),(3,3,3),(3,9)2 (3,27,27) 2 22314 1 1/4 5/2
<39,#1;3> <37,285>-#1;1 7 2 (3,3) H.4 (2122) (27,81),(3,3,3),(3,9)2 (3,27,27) 2 22314 1 1/4 5/2
<39,#1;4> <37,285>-#1;1 7 2 (3,3) E.6 (1122) (27,81),(3,3,3),(3,9)2 (3,27,27) 2 21314 1 0/3 0/0
<39,#1;5> <37,285>-#1;1 7 2 (3,3) E.14 (4122) (27,81),(3,3,3),(3,9)2 (3,27,27) 2 21314 1 0/3 0/0
<39,#1;6> <37,285>-#1;1 7 2 (3,3) E.14 (3122) (27,81),(3,3,3),(3,9)2 (3,27,27) 2 21314 1 0/3 0/0
<39,#1;7> <37,285>-#1;1 7 2 (9) c.18 (0122) (27,27),(3,3,3),(3,9)2 (3,9,27) 3 21312 1 0/3 0/0
<39,#1;1> <37,303>-#1;1 7 2 (3,3) c.21 (2034) (3,9),(27,81),(3,9)2 (3,27,27) 2 22314 1 1/4 5/2
<39,#1;2> <37,303>-#1;1 7 2 (3,3) E.8 (2234) (3,9),(27,81),(3,9)2 (3,27,27) 2 21314 1 0/3 0/0
<39,#1;3> <37,303>-#1;1 7 2 (3,3) G.16 (2134) (3,9),(27,81),(3,9)2 (3,27,27) 2 22314 1 1/4 5/2
<39,#1;4> <37,303>-#1;1 7 2 (3,3) E.9 (2434) (3,9),(27,81),(3,9)2 (3,27,27) 2 21314 1 0/3 0/0
<39,#1;5> <37,303>-#1;1 7 2 (3,3) G.16 (2134) (3,9),(27,81),(3,9)2 (3,27,27) 2 22314 1 1/4 5/2
<39,#1;6> <37,303>-#1;1 7 2 (3,3) E.9 (2334) (3,9),(27,81),(3,9)2 (3,27,27) 2 21314 1 0/3 0/0
<39,#1;7> <37,303>-#1;1 7 2 (3,3) c.21 (2034) (3,9),(27,27),(3,9)2 (3,9,27) 3 21312 1 0/3 0/0
<39,#1;2> <36,45>-#2;1 6 3 (3) H.4 (4111) (3,3,3)3,(3,9) (3,3,9) 3 22311 1 2/4 4/0;2/1
<39,#1;1> <36,49>-#2;1 6 3 (3,9) c.18 (0122) (27,27),(3,3,3),(3,9)2 (3,9,27) 3 22312 1 2/4 8/3;6/3
<39,#1;2> <36,49>-#2;2 6 3 (3,3) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 3 22312 1 2/4 4/1;2/1
<39,#1;2> <36,49>-#2;3 6 3 (3,3) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 3 22312 1 2/4 4/1;2/1
<39,#1;4> <36,54>-#2;1 6 3 (3,3) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 3 22312 1 2/4 4/1;2/1
<39,#1;1> <36,54>-#2;3 6 3 (3,9) c.21 (2034) (3,9),(27,27),(3,9)2 (3,9,27) 3 22312 1 2/4 8/3;6/3
<39,#1;4> <36,54>-#2;5 6 3 (3,3) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 3 22312 1 2/4 4/1;2/1
<39,#1;2> <36,57>-#2;1 6 3 (3) G.19 (2143) (3,9)4 (3,3,3,3) 3 23312 1 2/4 1/0;2/0
<39,#1;2> <36,57>-#2;5 6 3 (3) G.19 (2143) (3,9)4 (3,3,3,3) 3 23312 1 2/4 1/0;2/0
|G| = 59049
<310,#1;1> <36,45>-#2;1-#1;2 7 3 (3) H.4 (4111) (3,3,3)3,(3,9) (3,3,9) 3 22311 1 0/3 0/0
<310,#1;2> <36,45>-#2;1-#1;2 7 3 (3) H.4 (4111) (3,3,3)3,(3,9) (3,3,9) 3 22312 1 0/3 0/0
<310,#1;3> <36,45>-#2;1-#1;2 7 3 (3) H.4 (4111) (3,3,3)3,(3,9) (3,3,9) 3 21312 1 0/3 0/0
<310,#1;4> <36,45>-#2;1-#1;2 7 3 (3) H.4 (4111) (3,3,3)3,(3,9) (3,3,9) 3 21311 1 0/3 0/0
<310,#1;1> <36,49>-#2;1-#1;1 7 3 (27) c.18 (0122) (27,27),(3,3,3),(3,9)2 (3,9,27) 3 22312 1 0/3 0/0
<310,#1;2> <36,49>-#2;1-#1;1 7 3 (3,9) c.18 (0122) (27,81),(3,3,3),(3,9)2 (3,27,27) 3 22314 1 1/4 5/2
<310,#1;3> <36,49>-#2;1-#1;1 7 3 (3,9) H.4 (2122) (27,81),(3,3,3),(3,9)2 (3,27,27) 3 22314 1 1/4 5/2
<310,#1;4> <36,49>-#2;1-#1;1 7 3 (3,9) H.4 (2122) (27,81),(3,3,3),(3,9)2 (3,27,27) 3 22314 1 1/4 5/2
<310,#1;5> <36,49>-#2;1-#1;1 7 3 (3,9) E.6 (1122) (27,81),(3,3,3),(3,9)2 (3,27,27) 3 21314 1 0/3 0/0
<310,#1;6> <36,49>-#2;1-#1;1 7 3 (3,9) E.14 (4122) (27,81),(3,3,3),(3,9)2 (3,27,27) 3 21314 1 0/3 0/0
<310,#1;7> <36,49>-#2;1-#1;1 7 3 (3,9) E.14 (3122) (27,81),(3,3,3),(3,9)2 (3,27,27) 3 21314 1 0/3 0/0
<310,#1;8> <36,49>-#2;1-#1;1 7 3 (27) c.18 (0122) (27,27),(3,3,3),(3,9)2 (3,9,27) 3 21312 1 0/3 0/0
<310,#1;1> <36,49>-#2;2-#1;2 7 3 (9) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 3 22312 1 0/3 0/0
<310,#1;2> <36,49>-#2;2-#1;2 7 3 (3,3) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 3 22313 1 1/4 5/2
<310,#1;3> <36,49>-#2;2-#1;2 7 3 (3,3) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 3 21313 1 0/3 0/0
<310,#1;4> <36,49>-#2;2-#1;2 7 3 (9) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 3 21312 1 0/3 0/0
<310,#1;1> <36,49>-#2;3-#1;2 7 3 (9) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 3 22312 1 0/3 0/0
<310,#1;2> <36,49>-#2;3-#1;2 7 3 (3,3) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 3 22313 1 1/4 5/2
<310,#1;3> <36,49>-#2;3-#1;2 7 3 (3,3) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 3 21313 1 0/3 0/0
<310,#1;4> <36,49>-#2;3-#1;2 7 3 (9) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 3 21312 1 0/3 0/0
<310,#1;1> <36,54>-#2;1-#1;4 7 3 (3,3) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 3 21313 1 0/3 0/0
<310,#1;2> <36,54>-#2;1-#1;4 7 3 (3,3) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 3 22313 1 1/4 5/2
<310,#1;3> <36,54>-#2;1-#1;4 7 3 (9) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 3 22312 1 0/3 0/0
<310,#1;4> <36,54>-#2;1-#1;4 7 3 (9) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 3 21312 1 0/3 0/0
<310,#1;1> <36,54>-#2;3-#1;1 7 3 (3,9) c.21 (2034) (3,9),(27,81),(3,9)2 (3,27,27) 3 22314 1 1/4 5/2
<310,#1;2> <36,54>-#2;3-#1;1 7 3 (3,9) E.8 (2234) (3,9),(27,81),(3,9)2 (3,27,27) 3 21314 1 0/3 0/0
<310,#1;3> <36,54>-#2;3-#1;1 7 3 (3,9) G.16 (2134) (3,9),(27,81),(3,9)2 (3,27,27) 3 22314 1 1/4 5/2
<310,#1;4> <36,54>-#2;3-#1;1 7 3 (3,9) E.9 (2434) (3,9),(27,81),(3,9)2 (3,27,27) 3 21314 1 0/3 0/0
<310,#1;5> <36,54>-#2;3-#1;1 7 3 (3,9) G.16 (2134) (3,9),(27,81),(3,9)2 (3,27,27) 3 22314 1 1/4 5/2
<310,#1;6> <36,54>-#2;3-#1;1 7 3 (3,9) E.9 (2334) (3,9),(27,81),(3,9)2 (3,27,27) 3 21314 1 0/3 0/0
<310,#1;7> <36,54>-#2;3-#1;1 7 3 (27) c.21 (2034) (3,9),(27,27),(3,9)2 (3,9,27) 3 22312 1 0/3 0/0
<310,#1;8> <36,54>-#2;3-#1;1 7 3 (27) c.21 (2034) (3,9),(27,27),(3,9)2 (3,9,27) 3 21312 1 0/3 0/0
<310,#1;1> <36,54>-#2;5-#1;4 7 3 (3,3) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 3 21313 1 0/3 0/0
<310,#1;2> <36,54>-#2;5-#1;4 7 3 (3,3) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 3 22313 1 1/4 5/2
<310,#1;3> <36,54>-#2;5-#1;4 7 3 (9) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 3 22312 1 0/3 0/0
<310,#1;4> <36,54>-#2;5-#1;4 7 3 (9) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 3 21312 1 0/3 0/0
<310,#1;1> <36,57>-#2;1-#1;2 7 3 (3) G.19 (2143) (3,9)4 (3,3,3,3) 3 21312 1 0/3 0/0
<310,#1;1> <36,57>-#2;5-#1;2 7 3 (3) G.19 (2143) (3,9)4 (3,3,3,3) 3 21312 1 0/3 0/0
|G| = 177147
<311,#2;1> <36,45>-#2;1-#1;2 7 4 (3,3) H.4 (4111) (3,3,3)3,(3,9) (3,3,9) 3 22312 1 1/3 4/4
<311,#2;2> <36,45>-#2;1-#1;2 7 4 (3,3) H.4 (4111) a cover of <36,45> (3,3,3)3,(3,9) (3,3,9) 3 21312 1 0/2 0/0
<311,#2;1> <36,49>-#2;1-#1;1 7 4 (3,27) c.18 (0122) (27,81),(3,3,3),(3,9)2 (3,27,27) 3 22314 1 1/3 4/4
<311,#2;2> <36,49>-#2;1-#1;1 7 4 (3,27) H.4 (2122) (27,81),(3,3,3),(3,9)2 (3,27,27) 3 22314 1 1/3 4/4
<311,#2;3> <36,49>-#2;1-#1;1 7 4 (3,27) H.4 (2122) (27,81),(3,3,3),(3,9)2 (3,27,27) 3 22314 1 1/3 4/4
<311,#2;4> <36,49>-#2;1-#1;1 7 4 (3,27) E.6 (1122) the unique cover of <37,285>-#1;1-#1;4 (27,81),(3,3,3),(3,9)2 (3,27,27) 3 21314 1 0/2 0/0
<311,#2;5> <36,49>-#2;1-#1;1 7 4 (3,27) E.14 (4122) the unique cover of <37,285>-#1;1-#1;5 (27,81),(3,3,3),(3,9)2 (3,27,27) 3 21314 1 0/2 0/0
<311,#2;6> <36,49>-#2;1-#1;1 7 4 (3,27) E.14 (3122) the unique cover of <37,285>-#1;1-#1;6 (27,81),(3,3,3),(3,9)2 (3,27,27) 3 21314 1 0/2 0/0
<311,#2;1> <36,49>-#2;2-#1;2 7 4 (3,9) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 3 22313 1 1/3 4/4
<311,#2;2> <36,49>-#2;2-#1;2 7 4 (3,9) H.4 (2122) a cover of <37,286>-#1;2 (9,27),(3,3,3),(3,9)2 (3,9,27) 3 21313 1 0/2 0/0
<311,#2;1> <36,49>-#2;3-#1;2 7 4 (3,9) H.4 (2122) (9,27),(3,3,3),(3,9)2 (3,9,27) 3 22313 1 1/3 4/4
<311,#2;2> <36,49>-#2;3-#1;2 7 4 (3,9) H.4 (2122) a cover of <37,287>-#1;2 (9,27),(3,3,3),(3,9)2 (3,9,27) 3 21313 1 0/2 0/0
<311,#2;1> <36,54>-#2;1-#1;4 7 4 (3,9) G.16 (2134) a cover of <37,301>-#1;4 (3,9),(9,27),(3,9)2 (3,9,27) 3 21313 1 0/2 0/0
<311,#2;2> <36,54>-#2;1-#1;4 7 4 (3,9) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 3 22313 1 1/3 4/4
<311,#2;1> <36,54>-#2;3-#1;1 7 4 (3,27) c.21 (2034) (3,9),(27,81),(3,9)2 (3,27,27) 3 22314 1 1/3 4/4
<311,#2;2> <36,54>-#2;3-#1;1 7 4 (3,27) E.8 (2234) the unique cover of <37,303>-#1;1-#1;2 (3,9),(27,81),(3,9)2 (3,27,27) 3 21314 1 0/2 0/0
<311,#2;3> <36,54>-#2;3-#1;1 7 4 (3,27) G.16 (2134) (3,9),(27,81),(3,9)2 (3,27,27) 3 22314 1 1/3 4/4
<311,#2;4> <36,54>-#2;3-#1;1 7 4 (3,27) E.9 (2434) the unique cover of <37,303>-#1;1-#1;4 (3,9),(27,81),(3,9)2 (3,27,27) 3 21314 1 0/2 0/0
<311,#2;5> <36,54>-#2;3-#1;1 7 4 (3,27) G.16 (2134) (3,9),(27,81),(3,9)2 (3,27,27) 3 22314 1 1/3 4/4
<311,#2;6> <36,54>-#2;3-#1;1 7 4 (3,27) E.9 (2334) the unique cover of <37,303>-#1;1-#1;6 (3,9),(27,81),(3,9)2 (3,27,27) 3 21314 1 0/2 0/0
<311,#2;1> <36,54>-#2;5-#1;4 7 4 (3,9) G.16 (2134) a cover of <37,305>-#1;4 (3,9),(9,27),(3,9)2 (3,9,27) 3 21313 1 0/2 0/0
<311,#2;2> <36,54>-#2;5-#1;4 7 4 (3,9) G.16 (2134) (3,9),(9,27),(3,9)2 (3,9,27) 3 22313 1 1/3 4/4
<311,#2;1> <36,57>-#2;1-#1;2 7 4 (3,3) G.19 (2143) a cover of <36,57> (3,9)4 (3,3,3,3) 3 22313 1 0/2 0/0
<311,#2;2> <36,57>-#2;1-#1;2 7 4 (3,3) G.19 (2143) a cover of <36,57> (3,9)4 (3,3,3,3) 3 22313 1 0/2 0/0
<311,#2;1> <36,57>-#2;5-#1;2 7 4 (3,3) G.19 (2143) a cover of <36,57> (3,9)4 (3,3,3,3) 3 22313 1 0/2 0/0
<311,#2;2> <36,57>-#2;5-#1;2 7 4 (3,3) G.19 (2143) a cover of <36,57> (3,9)4 (3,3,3,3) 3 22313 1 0/2 0/0


1.2. Abelianization G/G' of Type (3,9):


We select two generators x,y of G = <x,y> such that
x9 and y3 are contained in G'.
Then we define the order of the maximal normal subgroups by
M1 = <x,Φ(G)>, M2 = <xy,Φ(G)>, M3 = <xy2,Φ(G)>, M4 = <y,Φ(G)>,
where Φ(G) = G3G' denotes the Frattini subgroup of G.
M4 is distinguished by a bicyclic quotient M4/G' ≅ (3,3),
whereas Mi/G' is cyclic of order 9 for 1 ≤ i ≤ 3.
This also defines the order of the components of (punctured) TTT and TKT.
For the pTKT, however, we must additionally specify the order of possible capitulants.
We indicate y by the symbol 1, x3y by 2, x3y2 by 3,
and x3 by the distinguished symbol 4, since Φ(G) = <x3,G'>.
Finally, the symbol 0 indicates the 2-dimensional capitulation of M4.


There are only 2 metabelian Schur σ-groups, <36,14> and <36,15>, which are of coclass 3.
Schur σ-groups G of derived length dl(G) = 3 start with order |G| = 39 and coclass cc(G) = 4,
forming Schur covers G of unbalanced σ-groups H ≅ G/G''.


G π(G) cl cc Z(G) TKT Mi TTT Φ(G)/Φ(G)' G'/G'' dl Aut σ Rank Desc
|G| = 27
<33,2> <1,1> 1 2 (3,9) a.1 (0000) <32,1>3,<32,2> (9)3,(3,3) (3) 1 1 2233 1 1/3 2/1
|G| = 81
<34,3> <33,2> 2 2 (3,3) a.1 (0000) <33,2>3,<33,5> (3,9)3,(3,3,3) (3,3) (3) 2 2235 1 3/4 9/5;18/16;4/4
<34,6> <33,2> 2 2 (9) A.1 (1111) <33,1>3,<33,2> (27)3,(3,9) (9) (3) 2 2134 0 0/2 0/0
|G| = 243
<35,13> = E <34,3> 3 2 (3,3) b.15 (0004) <34,3>3,<34,15>
<33,5>3,<33,5>
(3,9)3,(3,3,3,3)
(3,3,3)3,(3,3,3)
(3,3,3) (3,3) 2 2237 1 1/4 7/4
<35,14> = H <34,3> 3 2 (3,3) b.15 (0004) <34,3>3,<34,11>
<33,2>3,<33,5>
(3,9)3,(3,3,9)
(3,9)3,(3,3,3)
(3,3,3) (3,3) 2 2237 1 1/4 6/2
<35,15> = G <34,3> 3 2 (3,3) a.1 (0000) <34,3>3,<34,11> (3,9)3,(3,3,9) (3,3,3) (3,3) 2 2237 1 1/4 6/2
<35,16> = F <34,3> 3 2 (9) b.2 (0003) <34,6>3,<34,11> (3,9)3,(3,3,9) (3,9) (3,3) 2 2136 1 0/3 0/0
<35,17> = A <34,3> 3 2 (3,3) a.1 (0000) <34,3>2,<34,11>,<34,12> (3,9)2,(3,3,9),(3,3,3) (3,3,3) (3,3) 2 2236 1 1/4 7/2
<35,18> = D <34,3> 3 2 (3,3) b.16 (0040) <34,3>2,<34,11>,<34,13> (3,9)2,(3,3,9),(3,3,3) (3,3,3) (3,3) 2 2136 1 0/3 0/0
<35,19> = C <34,3> 3 2 (9) b.3 (0010) <34,6>2,<34,5>,<34,14> (3,9)2,(3,27),(3,3,3) (3,9) (3,3) 2 2235 1 0/3 0/0
<35,20> = B <34,3> 3 2 (9) b.3 (0010) <34,6>2,<34,5>,<34,14> (3,9)2,(3,27),(3,3,3) (3,9) (3,3) 2 2235 1 0/3 0/0
|G| = 729
<36,#1;1> = <36,65> <35,13> 4 2 (3) b.15 (0004) <35,13>3,<35,62>
<34,15>3,<34,15>
(3,9)3,(3,3,3,3)
(3,3,3,3)3,(3,3,3,3)
(3,3,3,3) (3,3,3) 2 2239 1 2/5 10/3;21/12
<36,#1;2> = <36,73> <35,14> 4 2 (3) b.15 (0004) <35,14>3,<35,34>
<34,2>3,<34,11>
(3,9)3,(3,3,9)
(9,9)3,(3,3,9)
(3,3,9) (3,9) 2 2239 1 1/4 1/0
<36,#1;2> = <36,79> <35,15> 4 2 (3,3) a.1 (0000) <35,15>3,<35,31> (3,9)3,(3,9,9) (3,3,9) (3,9) 2 2239 1 1/4 7/3
<36,#1;1> = <36,84> <35,17> 4 2 (3,3) a.1 (0000) <35,15>2,<35,31>,<35,53> (3,9)2,(3,9,9),(3,3,3) (3,3,9) (3,9) 2 2238 1 1/4 9/2
<36,9> <34,3> 3 3 (3,3,3) b.15 (0004) <35,32>3,<35,62> (3,3,9)3,(3,3,3,3) (3,3,3,3) (3,3,3) 2 2239 1 3/5 15/10;61/61;37/37
<36,10> <34,3> 3 3 (3,3,3) b.31 (0444) <35,32>4 (3,3,9)4 (3,3,3,3) (3,3,3) 2 2238 1 2/4 12/8;13/13
<36,11> <34,3> 3 3 (3,3,3) c.27 (0440) <35,32>4 (3,3,9)4 (3,3,3,3) (3,3,3) 2 2238 1 2/4 12/8;13/13
<36,12> <34,3> 3 3 (3,3,3) A.20 (4444) <35,32>3,<35,63> (3,3,9)3,(3,3,3,3) (3,3,3,3) (3,3,3) 2 2139 1 2/4 6/4;9/9
<36,13> <34,3> 3 3 (3,9) d.10 (0113) <35,49>,<35,12>2,<35,35> (3,3,9),(3,27)2,(3,3,9) (3,3,9) (3,3,3) 2 2137 1 1/3 5/5
<36,14> <34,3> 3 3 (3,9) D.11 (4233) <35,49>,<35,12>2,<35,36> (3,3,9),(3,27)2,(3,3,9) (3,3,9) (3,3,3) 2 2137 1 0/2 0/0
<36,15> <34,3> 3 3 (3,9) D.11 (4323) <35,49>,<35,12>2,<35,36> (3,3,9),(3,27)2,(3,3,9) (3,3,9) (3,3,3) 2 2137 1 0/2 0/0
<36,16> <34,3> 3 3 (3,9) B.7 (1114) <35,12>3,<35,64> (3,27)3,(3,3,3,3) (3,3,9) (3,3,3) 2 2238 1 1/3 4/4
<36,17> <34,3> 3 3 (3,9) E.12 (1234) <35,12>3,<35,34> (3,27)3,(3,3,9) (3,3,9) (3,3,3) 2 2238 1 1/3 3/3
<36,18> <34,3> 3 3 (3,9) e.14 (1320) <35,12>3,<35,34> (3,27)3,(3,3,9) (3,3,9) (3,3,3) 2 2238 1 1/3 3/3
<36,19> <34,3> 3 3 (3,9) B.7 (1114) <35,12>3,<35,64> (3,27)3,(3,3,3,3) (3,3,9) (3,3,3) 2 2238 1 1/3 4/4
<36,20> <34,3> 3 3 (3,9) E.12 (1324) <35,12>3,<35,34> (3,27)3,(3,3,9) (3,3,9) (3,3,3) 2 2238 1 1/3 3/3
<36,21> <34,3> 3 3 (3,9) e.14 (1230) <35,12>3,<35,34> (3,27)3,(3,3,9) (3,3,9) (3,3,3) 2 2238 1 1/3 3/3
|G| = 2187
<37,#1;1> = <37,319> <36,65> 5 2 (3) b.15 (0004) <36,65>3,<36,425>
<35,67>,<35,62>3
(3,9)3,(3,3,3,3)
(3,3,3,3,3),(3,3,3,3)3
(3,3,3,3) (3,3,3,3) 2 22310 1 1/5 16/6
<37,#1;2…#1;3> = <37,320…321> <36,65> 5 2 (3) b.15 (0004) <36,65>3,<36,428>
<35,61>,<35,63>2,<35,62>
(3,9)3,(3,3,3,3)
(3,3,3,9),(3,3,3,3)3
(3,3,3,3) (3,3,3,3) 2 22310 1 1/5 13/4
<37,#1;4> = <37,322> <36,65> 5 2 (3) b.15 (0004) <36,67>3,<36,426>
<35,67>,<35,63>3
(3,9)3,(3,3,3,3)
(3,3,3,3,3),(3,3,3,3)3
(3,3,3,3) (3,3,3,3) 2 21310 1 0/4 0/0
<37,#1;5> = <37,323> <36,65> 5 2 (3) b.15 (0004) <36,67>3,<36,428>
<35,61>,<35,63>,<35,62>,<35,63>
(3,9)3,(3,3,3,3)
(3,3,3,9),(3,3,3,3)3
(3,3,3,3) (3,3,3,3) 2 21310 1 0/4 0/0
<37,#1;6> = <37,324> <36,65> 5 2 (3) b.15 (0004) <36,67>3,<36,428>
<35,61>,<35,62>,<35,63>,<35,63>
(3,9)3,(3,3,3,3)
(3,3,3,9),(3,3,3,3)3
(3,3,3,3) (3,3,3,3) 2 21310 1 0/4 0/0
<37,#1;1> = <37,349> <36,73> 5 2 (3) b.15 (0004) <36,73>,<36,74>2,<36,273>
<35,10>,<35,11>2,<35,34>
(3,9)3,(3,3,9)
(9,27),(9,9)2,(3,3,9)
(3,3,9) (9,9) 2 2238 1 0/3 0/0
<37,#1;1> <36,79> 5 2 (3,3) a.1 (0000) <36,79>3,<36,238> (3,9)3,(3,9,27) (3,9,9) (9,9) 2 22311 1 1/4 6/2
<37,#1;2…#1;3> <36,79> 5 2 (3,3) b.15 (0004) <36,79>3,<36,238> (3,9)3,(3,9,27) (3,9,9) (9,9) 2 22311 1 1/4 6/2
<37,#1;4> <36,79> 5 2 (9) b.2 (0001) <36,80>3,<36,238> (3,9)3,(3,9,27) (3,9,9) (9,9) 2 21310 1 0/3 0/0
<37,#1;5> <36,79> 5 2 (3,3) a.1 (0000) <36,79>3,<36,239> (3,9)3,(3,9,9) (3,9,9) (9,9) 2 21310 1 0/3 0/0
<37,#1;6…#1;7> <36,79> 5 2 (9) a.1 (0000) <36,80>3,<36,239> (3,9)3,(3,9,9) (3,9,9) (9,9) 2 21310 1 0/3 0/0
<37,#1;1> <36,84> 5 2 (3,3) a.1 (0000) <36,79>2,<36,238>,<36,395> (3,9)2,(3,9,27),(3,3,3) (3,9,9) (9,9) 2 22310 1 1/4 7/2
<37,#1;2> <36,84> 5 2 (3,3) b.16 (0040) <36,79>2,<36,238>,<36,396> (3,9)2,(3,9,27),(3,3,3) (3,9,9) (9,9) 2 21310 1 0/3 0/0
<37,#1;3> <36,84> 5 2 (9) b.3 (0010) <36,80>2,<36,102>,<36,397> (3,9)2,(9,9,9),(3,3,3) (3,9,9) (9,9) 2 2239 1 0/3 0/0
<37,#1;4> <36,84> 5 2 (9) b.3 (0010) <36,80>2,<36,238>,<36,397> (3,9)2,(3,9,27),(3,3,3) (3,9,9) (9,9) 2 2239 1 0/3 0/0
<37,#1;5> <36,84> 5 2 (3,3) a.1 (0000) <36,79>2,<36,239>,<36,395> (3,9)2,(3,9,9),(3,3,3) (3,9,9) (9,9) 2 21310 1 1/4 8/2
<37,#1;6…#1;7> <36,84> 5 2 (3,3) a.1 (0000) <36,79>2,<36,239>,<36,396> (3,9)2,(3,9,9),(3,3,3) (3,9,9) (9,9) 2 21310 1 0/3 0/0
<37,#1;8> <36,84> 5 2 (9) a.1 (0000) <36,80>2,<36,104>,<36,397> (3,9)2,(3,9,9),(3,3,3) (3,9,9) (9,9) 2 2139 1 0/3 0/0
<37,#1;9> <36,84> 5 2 (9) a.1 (0000) <36,80>2,<36,239>,<36,397> (3,9)2,(3,9,9),(3,3,3) (3,9,9) (9,9) 2 2139 1 0/3 0/0
<37,#1;1> = <37,168> <36,13> 4 3 (3,9) d.10 (0113) <36,239>,<36,6>2,<36,278> (3,9,9),(3,27)2,(3,3,9) (3,9,9) (3,3,9) 2 2139 1 2/4 12/3;9/3
<37,#1;1> = <37,173> <36,16> 4 3 (3,3) B.7 (1114) <36,4>3,<36,428> (3,27)3,(3,3,3,3) (3,3,3,9) (3,3,3,3) 2 22310 1 2/4 8/6;9/9
<37,#1;2> = <37,178> <36,17> 4 3 (3,3) E.12 (1234) <36,5>3,<36,110> (3,27)3,(3,3,9) (3,9,9) (3,3,9) 2 22310 1 2/4 3/1;2/1
<37,#1;2> = <37,181> <36,18> 4 3 (3,9) e.14 (1320) <36,6>3,<36,104> (3,27)3,(3,9,9) (3,9,9) (3,3,9) 2 22310 1 2/4 5/3;4/3
<37,#1;1> = <37,183> <36,19> 4 3 (3,3) B.7 (1114) <36,4>3,<36,428> (3,27)3,(3,3,3,3) (3,3,3,9) (3,3,3,3) 2 22310 1 2/4 8/6;9/9
<37,#1;2> = <37,188> <36,20> 4 3 (3,3) E.12 (1324) <36,5>3,<36,112> (3,27)3,(3,3,9) (3,9,9) (3,3,9) 2 22310 1 2/4 3/1;2/1
<37,#1;2> = <37,191> <36,21> 4 3 (3,9) e.14 (1230) <36,6>3,<36,104> (3,27)3,(3,9,9) (3,9,9) (3,3,9) 2 22310 1 2/4 5/3;4/3
G π(G) cl cc Z(G) TKT Schur cover of TTT Φ(G)/Φ(G)' G'/G'' dl Aut σ Rank Desc
|G| = 6561
<38,#1;1> <37,319> 6 2 (3) b.15 (0004) <36,504>,<36,425>3 (3,9)3,(3,3,3,3)
(3,3,3,3,3,3),(3,3,3,3)3
(3,3,3,3) (3,3,3,3,3) 2 22312 1 2/6 30/11;87/87
<38,#1;2> <37,320…321> 6 2 (3) b.15 (0004) <36,415>,<36,431>2,<36,428> (3,9)3,(3,3,3,3)
(3,3,9,9),(3,3,3,3)3
(3,3,3,3) (3,3,3,9) 2 22312 1 1/5 3/0
<38,#1;1> <37,168> 5 3 (3,9) B.2 (1113) (9,9,9),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 2 21311 1 1/4 6/2
<38,#1;2> <37,168> 5 3 (3,9) C.4 (3113) (9,9,9),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 2 21311 1 0/3 0/0
<38,#1;3> <37,168> 5 3 (3,9) D.5 (2113) (9,9,9),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 2 21311 1 0/3 0/0
<38,#1;4> <37,168> 5 3 (3,9) B.2 (1113) (3,9,27),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 2 21311 1 1/4 6/2
<38,#1;5> <37,168> 5 3 (3,9) D.5 (2113) (3,9,27),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 2 21311 1 0/3 0/0
<38,#1;6> <37,168> 5 3 (3,9) C.4 (3113) (3,9,27),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 2 21311 1 0/3 0/0
<38,#1;7> <37,168> 5 3 (3,9) d.10 (0113) (3,9,27),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 2 21311 1 1/4 6/2
<38,#1;8> <37,168> 5 3 (3,9) D.10 (4113) (3,9,27),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 2 21311 1 0/3 0/0
<38,#1;9> <37,168> 5 3 (3,9) D.10 (4113) (3,9,27),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 2 21311 1 0/3 0/0
<38,#1;10…#1;12> <37,168> 5 3 (27) d.10 (0113) (3,9,9),(3,27)2,(3,3,9) (3,9,9) (3,3,9) 3 2139 1 0/3 0/0
<38,#1;1> <37,178> 5 3 (3,3) E.12 (1234) (3,27)3,(3,3,9) (3,9,9) (3,9,9) 2 22311 1 1/4 4/2
<38,#1;2> <37,178> 5 3 (3,3) E.12 (1234) (3,27)3,(3,3,9) (3,9,9) (3,9,9) 2 22311 1 0/3 0/0
<38,#1;3> <37,178> 5 3 (9) E.12 (1234) (3,27)3,(3,3,9) (3,9,9) (3,3,9) 3 2239 1 0/3 0/0
<38,#1;1> <37,181> 5 3 (3,9) e.14 (1320) (3,27)3,(3,9,27) (9,9,9) (3,9,9) 2 22312 1 1/4 4/2
<38,#1;2…#1;3> <37,181> 5 3 (3,9) E.12 (1324) (3,27)3,(3,9,27) (9,9,9) (3,9,9) 2 22312 1 1/4 4/2
<38,#1;4> <37,181> 5 3 (3,9) D.6 (1321) (3,27)3,(3,9,27) (9,9,9) (3,9,9) 2 21311 1 0/3 0/0
<38,#1;5> <37,181> 5 3 (27) e.14 (1320) (3,27)3,(3,9,9) (3,9,9) (3,3,9) 3 2239 1 0/3 0/0
<38,#1;1> <37,188> 5 3 (3,3) E.12 (1324) (3,27)3,(3,3,9) (3,9,9) (3,9,9) 2 22311 1 1/4 4/2
<38,#1;2> <37,188> 5 3 (3,3) E.12 (1324) (3,27)3,(3,3,9) (3,9,9) (3,9,9) 2 21311 1 0/3 0/0
<38,#1;3> <37,188> 5 3 (9) E.12 (1324) (3,27)3,(3,3,9) (3,9,9) (3,3,9) 3 2239 1 0/3 0/0
<38,#1;1> <37,191> 5 3 (3,9) e.14 (1230) (3,27)3,(3,9,27) (9,9,9) (3,9,9) 2 22312 1 1/4 4/2
<38,#1;2…#1;3> <37,191> 5 3 (3,9) E.12 (1234) (3,27)3,(3,9,27) (9,9,9) (3,9,9) 2 22312 1 1/4 4/2
<38,#1;4> <37,191> 5 3 (3,9) D.6 (1231) (3,27)3,(3,9,27) (9,9,9) (3,9,9) 2 21311 1 0/3 0/0
<38,#1;5> <37,191> 5 3 (27) e.14 (1230) (3,27)3,(3,9,9) (3,9,9) (3,3,9) 3 2239 1 0/3 0/0
|G| = 19683
<39,#1;1> <37,319>-#1;1 7 2 (3) b.15 (0004) (3,9)3,(3,3,3,3)
(3,3,3,3,3,3,3),(3,3,3,3)3
(3,3,3,3) (3,3,3,3,3,3) 2 22314 1 1/6 22/8
<39,#1;1> <37,320…321>-#1;2 7 2 (3) b.15 (0004) (3,9)3,(3,3,3,3)
(3,9,9,9),(3,3,3,3)3
(3,3,3,3) (3,3,9,9) 2 22312 1 0/4 0/0
<39,#1;2> <37,320…321>-#1;2 7 2 (3) b.15 (0004) (3,9)3,(3,3,3,3)
(3,3,9,9),(3,3,3,3)3
(3,3,3,3) (3,3,9,9) 2 21312 1 0/4 0/0
<39,#1;4> <37,168>-#1;7 6 3 (3,9) d.10 (0113) (3,27,27),(3,27)2,(3,3,9) (9,9,27) (3,9,27) 2 21313 1 1/4 11/3
<39,#1;1> <37,181>-#1;1 6 3 (3,9) e.14 (1320) (3,27)3,(3,27,27) (9,9,27) (3,9,27) 2 22314 1 1/4 5/3
<39,#1;1> <37,191>-#1;1 6 3 (3,9) e.14 (1320) (3,27)3,(3,27,27) (9,9,27) (3,9,27) 2 22314 1 1/4 5/3
<39,#2;1> <37,168> 5 4 (3,27) B.2 (1113) (9,9,9),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 3 21311 1 1/3 5/5
<39,#2;2> <37,168> 5 4 (3,27) C.4 (3113) the unique cover of <37,168>-#1;2 (9,9,9),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 3 21311 1 0/2 0/0
<39,#2;3> <37,168> 5 4 (3,27) D.5 (2113) the unique cover of <37,168>-#1;3 (9,9,9),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 3 21311 1 0/2 0/0
<39,#2;4> <37,168> 5 4 (3,27) B.2 (1113) (3,9,27),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 3 21311 1 1/3 5/5
<39,#2;5> <37,168> 5 4 (3,27) D.5 (2113) the unique cover of <37,168>-#1;5 (3,9,27),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 3 21311 1 0/2 0/0
<39,#2;6> <37,168> 5 4 (3,27) C.4 (3113) the unique cover of <37,168>-#1;6 (3,9,27),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 3 21311 1 0/2 0/0
<39,#2;7> <37,168> 5 4 (3,27) d.10 (0113) (3,9,27),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 3 21311 1 1/3 5/5
<39,#2;8> <37,168> 5 4 (3,27) D.10 (4113) the unique cover of <37,168>-#1;8 (3,9,27),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 3 21311 1 0/2 0/0
<39,#2;9> <37,168> 5 4 (3,27) D.10 (4113) the unique cover of <37,168>-#1;9 (3,9,27),(3,27)2,(3,3,9) (9,9,9) (3,9,9) 3 21311 1 0/2 0/0
<39,#2;1> <37,178> 5 4 (3,9) E.12 (1234) (3,27)3,(3,3,9) (3,9,9) (3,9,9) 3 22311 1 1/3 3/3
<39,#2;2> <37,178> 5 4 (3,9) E.12 (1234) a cover of <37,178>-#1;2 (3,27)3,(3,3,9) (3,9,9) (3,9,9) 3 21311 1 0/2 0/0
<39,#2;1> <37,181> 5 4 (3,27) e.14 (1320) (3,27)3,(3,9,27) (9,9,9) (3,9,9) 3 22312 1 1/3 3/3
<39,#2;2…#2;3> <37,181> 5 4 (3,27) E.12 (1324) (3,27)3,(3,9,27) (9,9,9) (3,9,9) 3 22312 1 1/3 3/3
<39,#2;4> <37,181> 5 4 (3,27) D.6 (1321) the unique cover of <37,181>-#1;4 (3,27)3,(3,9,27) (9,9,9) (3,9,9) 3 21311 1 0/2 0/0
<39,#2;1> <37,188> 5 4 (3,9) E.12 (1324) (3,27)3,(3,3,9) (3,9,9) (3,9,9) 3 22311 1 1/3 3/3
<39,#2;2> <37,188> 5 4 (3,9) E.12 (1324) a cover of <37,188>-#1;2 (3,27)3,(3,3,9) (3,9,9) (3,9,9) 3 21311 1 0/2 0/0
<39,#2;1> <37,191> 5 4 (3,27) e.14 (1230) (3,27)3,(3,9,27) (9,9,9) (3,9,9) 3 22312 1 1/3 3/3
<39,#2;2…#2;3> <37,191> 5 4 (3,27) E.12 (1234) (3,27)3,(3,9,27) (9,9,9) (3,9,9) 3 22312 1 1/3 3/3
<39,#2;4> <37,191> 5 4 (3,27) D.6 (1231) the unique cover of <37,191>-#1;4 (3,27)3,(3,9,27) (9,9,9) (3,9,9) 3 21311 1 0/2 0/0
|G| = 59049
<310,#1;1> <37,168>-#1;7-#1;4 7 3 (3,9) B.2 (1113) (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 2 21315 1 1/4 6/2
<310,#1;2> <37,168>-#1;7-#1;4 7 3 (3,9) D.5 (2113) (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 2 21315 1 0/3 0/0
<310,#1;3> <37,168>-#1;7-#1;4 7 3 (3,9) C.4 (3113) (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 2 21315 1 0/3 0/0
<310,#1;4> <37,168>-#1;7-#1;4 7 3 (3,9) d.10 (0113) (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 2 21315 1 1/4 6/2
<310,#1;5> <37,168>-#1;7-#1;4 7 3 (3,9) D.10 (4113) (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 2 21315 1 0/3 0/0
<310,#1;6> <37,168>-#1;7-#1;4 7 3 (3,9) D.10 (4113) (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 2 21315 1 0/3 0/0
<310,#1;7> <37,168>-#1;7-#1;4 7 3 (3,9) B.2 (1113) (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 2 21315 1 1/4 6/2
<310,#1;8> <37,168>-#1;7-#1;4 7 3 (3,9) C.4 (3113) (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 2 21315 1 0/3 0/0
<310,#1;9> <37,168>-#1;7-#1;4 7 3 (3,9) D.5 (2113) (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 2 21315 1 0/3 0/0
<310,#1;10…#1;11> <37,168>-#1;7-#1;4 7 3 (27) d.10 (0113) (3,27,27),(3,27)2,(3,3,9) (9,9,27) (3,9,27) 3 21313 1 0/3 0/0
<310,#1;1> <37,181>-#1;1-#1;1 7 3 (3,9) e.14 (1320) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 2 22316 1 1/4 4/2
<310,#1;2…#1;3> <37,181>-#1;1-#1;1 7 3 (3,9) E.12 (1324) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 2 22316 1 1/4 4/2
<310,#1;4> <37,181>-#1;1-#1;1 7 3 (3,9) D.6 (1321) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 2 22316 1 0/3 0/0
<310,#1;5> <37,181>-#1;1-#1;1 7 3 (27) e.14 (1320) (3,27)3,(3,27,27) (9,9,27) (3,9,27) 3 21314 1 0/3 0/0
<310,#1;1> <37,191>-#1;1-#1;1 7 3 (3,9) e.14 (1320) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 2 22316 1 1/4 4/2
<310,#1;2…#1;3> <37,191>-#1;1-#1;1 7 3 (3,9) E.12 (1324) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 2 22316 1 1/4 4/2
<310,#1;4> <37,191>-#1;1-#1;1 7 3 (3,9) D.6 (1321) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 2 22316 1 0/3 0/0
<310,#1;5> <37,191>-#1;1-#1;1 7 3 (27) e.14 (1320) (3,27)3,(3,27,27) (9,9,27) (3,9,27) 3 21314 1 0/3 0/0
<310,#1;4> <37,168>-#2;7 6 4 (3,27) d.10 (0113) (3,27,27),(3,27)2,(3,3,9) (9,9,27) (3,9,27) 3 21313 1 2/4 12/3;9/3
<310,#1;1> <37,181>-#2;1 6 4 (3,27) e.14 (1230) (3,27)3,(3,27,27) (9,9,27) (3,9,27) 3 22314 1 2/4 5/3;4/3
<310,#1;1> <37,191>-#2;1 6 4 (3,27) e.14 (1230) (3,27)3,(3,27,27) (9,9,27) (3,9,27) 3 22314 1 2/4 5/3;4/3
|G| = 177147
<311,#1;1> <37,181>-#2;1-#1;1 7 4 (3,27) e.14 (1230) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 3 22316 1 1/4 4/2
<311,#1;2…#1;3> <37,181>-#2;1-#1;1 7 4 (3,27) E.12 (1234) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 3 22316 1 1/4 4/2
<311,#1;4> <37,181>-#2;1-#1;1 7 4 (3,27) D.6 (1231) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 3 21315 1 0/3 0/0
<311,#1;5> <37,181>-#2;1-#1;1 7 4 (81) e.14 (1230) (3,27)3,(3,27,27) (9,9,27) (3,9,27) 3 22313 1 0/3 0/0
<311,#1;1> <37,191>-#2;1-#1;1 7 4 (3,27) e.14 (1230) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 3 22316 1 1/4 4/2
<311,#1;2…#1;3> <37,191>-#2;1-#1;1 7 4 (3,27) E.12 (1234) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 3 22316 1 1/4 4/2
<311,#1;4> <37,191>-#2;1-#1;1 7 4 (3,27) D.6 (1231) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 3 21315 1 0/3 0/0
<311,#1;5> <37,191>-#2;1-#1;1 7 4 (81) e.14 (1230) (3,27)3,(3,27,27) (9,9,27) (3,9,27) 3 22313 1 0/3 0/0
|G| = 531441
<312,#2;1> <37,168>-#2;7-#1;4 7 5 (3,81) B.2 (1113) (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 3 21315 1 1/3 5/5
<312,#2;2> <37,168>-#2;7-#1;4 7 5 (3,81) D.5 (2113) the unique cover of <37,168>-#1;7-#1;4-#1;2 (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 3 21315 1 0/2 0/0
<312,#2;3> <37,168>-#2;7-#1;4 7 5 (3,81) C.4 (3113) the unique cover of <37,168>-#1;7-#1;4-#1;3 (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 3 21315 1 0/2 0/0
<312,#2;4> <37,168>-#2;7-#1;4 7 5 (3,81) d.10 (0113) (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 3 21315 1 1/3 5/5
<312,#2;5> <37,168>-#2;7-#1;4 7 5 (3,81) D.10 (4113) the unique cover of <37,168>-#1;7-#1;4-#1;5 (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 3 21315 1 0/2 0/0
<312,#2;6> <37,168>-#2;7-#1;4 7 5 (3,81) D.10 (4113) the unique cover of <37,168>-#1;7-#1;4-#1;6 (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 3 21315 1 0/2 0/0
<312,#2;7> <37,168>-#2;7-#1;4 7 5 (3,81) B.2 (1113) (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 3 21315 1 1/3 5/5
<312,#2;8> <37,168>-#2;7-#1;4 7 5 (3,81) C.4 (3113) the unique cover of <37,168>-#1;7-#1;4-#1;8 (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 3 21315 1 0/2 0/0
<312,#2;9> <37,168>-#2;7-#1;4 7 5 (3,81) D.5 (2113) the unique cover of <37,168>-#1;7-#1;4-#1;9 (3,27,81),(3,27)2,(3,3,9) (9,27,27) (3,27,27) 3 21315 1 0/2 0/0
<312,#2;1> <37,181>-#2;1-#1;1 7 5 (3,81) e.14 (1230) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 3 22316 1 1/3 3/3
<312,#2;2…#2;3> <37,181>-#2;1-#1;1 7 5 (3,81) E.12 (1234) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 3 22316 1 1/3 3/3
<312,#2;4> <37,181>-#2;1-#1;1 7 5 (3,81) D.6 (1231) the unique cover of <37,181>-#1;1-#1;1-#1;4 (3,27)3,(3,27,81) (9,27,27) (3,27,27) 3 21315 1 0/2 0/0
<312,#2;1> <37,191>-#2;1-#1;1 7 5 (3,81) e.14 (1230) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 3 22316 1 1/3 3/3
<312,#2;2…#2;3> <37,191>-#2;1-#1;1 7 5 (3,81) E.12 (1234) (3,27)3,(3,27,81) (9,27,27) (3,27,27) 3 22316 1 1/3 3/3
<312,#2;4> <37,191>-#2;1-#1;1 7 5 (3,81) D.6 (1231) the unique cover of <37,191>-#1;1-#1;1-#1;4 (3,27)3,(3,27,81) (9,27,27) (3,27,27) 3 21315 1 0/2 0/0


2. Coclass Trees of 5-Groups G:


2.1. Abelianization G/G' of Type (5,5):


Root of coclass graph G(5,1) is the 5-elementary bicyclic group
<52,2> ≅ (5,5), which is not coclass settled.
Exponent-5 descendant is the abelian group <53,2> ≅ (5,25).
Generalized descendants of depth 2 are
the abelian group <54,2> ≅ (25,25)
and the groups <54,3|4> with abelianization (5,25).
Generalized descendant of depth 3 is the group <55,2> with abelianization (25,25).


It is well known that the unique tree of coclass graph G(5,1) has branches of unbounded depth
and entirely consists of groups G with G/G' ≅ (5,5) and bounded derived length.
We show that the smallest Schur+1 σ-groups occur on branch B(3) of this tree
but then we leave G(5,1) and proceed to groups G with cc(G) ≥ 2 and G/G' ≅ (5,5).


There are 6 metabelian Schur σ-groups, <55,8…9> and <55,11…14>, which are of coclass 2.
Schur σ-groups G of derived length dl(G) = 3 start with order |G| = 58 and coclass cc(G) = 3,
forming Schur covers G of unbalanced σ-groups H ≅ G/G''.


G π(G) cl cc Z(G) TKT Mi TTT G'/G'' dl Aut σ Rank Desc
|G| = 25
<52,2> <1,1> 1 1 (5,5) (000000) <51,1>6 (5)6 1 1 253151 1 3/3 3/2;3/3;1/1
|G| = 125
<53,3> <52,2> 2 1 (5) (000000) <52,2>6 (5,5)6 (5) 2 253153 1 2/4 4/1;12/6
<53,4> <52,2> 2 1 (5) (111111) <52,2>,<52,1>5 (5,5),(25)5 (5) 2 2253 0 0/2 0/0
|G| = 625
<54,7> <53,3> 3 1 (5) (000000) <53,5>,<53,3>5 (5,5,5),(5,5)5 (5,5) 2 2455 1 1/4 9/2
<54,8> <53,3> 3 1 (5) (100000) <53,5>,<53,4>5 (5,5,5),(5,5)5 (5,5) 2 2255 1 0/3 0/0
<54,9 | 10> <53,3> 3 1 (5) (200000) <53,2>,<53,3>,<53,4>4 (5,25),(5,5)5 (5,5) 2 2354 1 0/3 0/0
|G| = 3125
<55,3> <53,3> 3 2 (5,5) (000000) <54,12>6 (5,5,5)6 (5,5,5) 2 253157 1 3/5 8/3;61/61;47/47
<55,4> <53,3> 3 2 (5,5) (101111) <54,12>,<54,13>,<54,3>4 (5,5,5)2,(5,25)4 (5,5,5) 2 2256 1 1/3 7/7
<55,5 | 6> <53,3> 3 2 (5,5) (202222) <54,3>,<54,12>,<54,3>4 (5,25),(5,5,5),(5,25)4 (5,5,5) 2 2357 1 1/3 4/4
<55,7> <53,3> 3 2 (5,5) (213546) <54,3>2,<54,13>,<54,3>2,<54,13> (5,25)2,(5,5,5),(5,25)2,(5,5,5) (5,5,5) 2 2356 1 1/3 6/6
<55,8> <53,3> 3 2 (5,5) (612435) <54,3>3,<54,13>,<54,3>2 (5,25)3,(5,5,5),(5,25)2 (5,5,5) 2 2157 1 0/2 0/0
<55,9> <53,3> 3 2 (5,5) (516324) <54,3>6 (5,25)6 (5,5,5) 2 213156 1 0/2 0/0
<55,10> <53,3> 3 2 (5,5) (215634) <54,3>6 (5,25)6 (5,5,5) 2 223156 1 1/3 5/5
<55,11> <53,3> 3 2 (5,5) (412356) <54,3>4,<54,13>2 (5,25)4,(5,5,5)2 (5,5,5) 2 2256 1 0/2 0/0
<55,12> <53,3> 3 2 (5,5) (614523) <54,3>6 (5,25)6 (5,5,5) 2 213156 1 0/2 0/0
<55,13> <53,3> 3 2 (5,5) (612435) <54,3>3,<54,13>,<54,3>2 (5,25)3,(5,5,5),(5,25)2 (5,5,5) 2 2157 1 0/2 0/0
<55,14> <53,3> 3 2 (5,5) (123456) <54,13>6 (5,5,5)6 (5,5,5) 2 233157 1 0/2 0/0
|G| = 15625
<56,#1;1> <55,3> 4 2 (5,5) (000000) <55,72>,<55,61>5 (5,5,5,5),(5,5,5)5 (5,5,5,5) 2 2459 1 2/6 71/3;5010/421
<56,#1;2> <55,3> 4 2 (5,5) (100000) <55,72>,<55,62>5 (5,5,5,5),(5,5,5)5 (5,5,5,5) 2 2259 0 0/4 0/0
<56,#1;3> <55,3> 4 2 (5,5) (200000) <55,40>,<55,61>,<55,62>4 (5,5,25),(5,5,5)5 (5,5,5,5) 2 2258 0 0/4 0/0
<56,#1;4> <55,3> 4 2 (5) (000000) <55,45>,<55,61>2,<55,45>,<55,61>2 (5,5,5)6 (5,5,5,5) 2 2558 1 2/6 149/0;12395/124
<56,#1;5> <55,3> 4 2 (5) (000000) <55,45>,<55,62>2,<55,46>,<55,62>2 (5,5,5)6 (5,5,5,5) 2 2258 0 0/4 0/0
<56,#1;6> <55,3> 4 2 (5) (000000) <55,46>,<55,61>,<55,62>,<55,46>,<55,62>2 (5,5,5)6 (5,5,5,5) 2 2158 0 0/4 0/0
<56,#1;7> <55,3> 4 2 (5) (000000) <55,61>6 (5,5,5)6 (5,5,5,5) 2 243158 1 2/6 95/0;8296/88
<56,#1;8> <55,3> 4 2 (5) (000000) <55,61>,<55,62>5 (5,5,5)6 (5,5,5,5) 2 2158 0 0/4 0/0
<56,#1;1> = <56,564> <55,4> 4 2 (5,5) (101111) <55,61>,<55,73>,<55,18>4 (5,5,5),(5,5,5,5),(5,25)4 (5,5,5,5) 2 2258 1 2/4 30/5;75/75
<56,#1;2> <55,4> 4 2 (5,5) (121111) <55,62>,<55,73>,<55,18>4 (5,5,5),(5,5,5,5),(5,25)4 (5,5,5,5) 2 58 0 1/3 5/0
<56,#1;3> <55,4> 4 2 (5,5) (111111) <55,61>,<55,40>,<55,18>4 (5,5,5),(5,5,25),(5,25)4 (5,5,5,5) 2 58 0 2/4 50/5;125/125
<56,#1;4…#1;7> <55,4> 4 2 (5,5) (131111) <55,62>,<55,40>,<55,18>4 (5,5,5),(5,5,25),(5,25)4 (5,5,5,5) 2 58 0 1/3 5/0
<56,#1;1> = <56,557> <55,5> 4 2 (5,5) (202222) <55,18>,<55,72>,<55,18>4 (5,25),(5,5,5,5),(5,25)4 (5,5,5,5) 2 2359 1 2/4 8/6;10/10
<56,#1;2…#1;3> <55,5> 4 2 (5,5) (222222) <55,18>,<55,72>,<55,18>4 (5,25),(5,5,5,5),(5,25)4 (5,5,5,5) 2 2259 0 2/4 8/6;10/10
<56,#1;4> <55,5> 4 2 (5,5) (212222) <55,18>,<55,40>,<55,18>4 (5,25),(5,5,25),(5,25)4 (5,5,5,5) 2 2158 0 1/3 15/15
<56,#1;1> = <56,558> <55,6> 4 2 (5,5) (202222) <55,18>,<55,72>,<55,18>4 (5,25),(5,5,5,5),(5,25)4 (5,5,5,5) 2 2359 1 2/4 8/6;10/10
<56,#1;2> <55,6> 4 2 (5,5) (222222) <55,18>,<55,72>,<55,18>4 (5,25),(5,5,5,5),(5,25)4 (5,5,5,5) 2 2259 0 2/4 8/6;15/15
<56,#1;3> <55,6> 4 2 (5,5) (222222) <55,18>,<55,72>,<55,18>4 (5,25),(5,5,5,5),(5,25)4 (5,5,5,5) 2 2259 0 2/4 8/6;10/10
<56,#1;4> <55,6> 4 2 (5,5) (212222) <55,18>,<55,40>,<55,18>4 (5,25),(5,5,25),(5,25)4 (5,5,5,5) 2 2158 0 1/3 15/15
<56,#1;1> <55,7> 4 2 (5) (213546) <55,19>2,<55,50>,<55,20>2,<55,50> (5,25)2,(5,5,5),(5,25)2,(5,5,5) (5,5,25) 2 58 0 1/3 1/0
<56,#1;2…#1;5> <55,7> 4 2 (5) (213546) <55,19>2,<55,50>,<55,20>2,<55,50> (5,25)2,(5,5,5),(5,25)2,(5,5,5) (5,5,25) 2 2158 0 1/3 1/0
<56,#1;6> = <56,674> <55,7> 4 2 (5) (213546) <55,19>2,<55,50>,<55,20>2,<55,50> (5,25)2,(5,5,5),(5,25)2,(5,5,5) (5,5,25) 2 2358 1 2/4 3/0;3/1
<56,#1;1…#1;4> <55,10> 4 2 (5) (215634) <55,19>,<55,20>2,<55,19>2,<55,20> (5,25)6 (5,5,25) 2 2158 0 1/3 1/0
<56,#1;5> = <56,680> <55,10> 4 2 (5) (215634) <55,19>,<55,20>2,<55,19>2,<55,20> (5,25)6 (5,5,25) 2 223158 1 2/4 2/0;3/1
G π(G) cl cc Z(G) TKT Schur cover of TTT G'/G'' dl Aut σ Rank Desc
|G| = 78125
<57,#1;1> <56,564> 5 2 (5,5) (111111) (5,5,5),(5,5,5,5,5),(5,25)4 (5,5,5,5,5) 2 22510 1 1/4 12/8
<57,#1;2…#1;3> <56,564> 5 2 (5,5) (131111) (5,5,5),(5,5,5,5,5),(5,25)4 (5,5,5,5,5) 2 21510 1 0/3 0/0
<57,#1;4> <56,564> 5 2 (5,5) (101111) (5,5,5),(5,5,5,25),(5,25)4 (5,5,5,5,5) 2 22510 1 1/4 12/8
<57,#1;5…#1;6> <56,564> 5 2 (5,5) (121111) (5,5,5),(5,5,5,25),(5,25)4 (5,5,5,5,5) 2 21510 1 0/3 0/0
<57,#1;7 | #1;10 | #1;13> <56,564> 5 2 (5,5) (111111) (5,5,5),(5,5,5,25),(5,25)4 (5,5,5,5,5) 2 22510 1 1/4 12/8
<57,#1;8…#1;9 | #1;11…#1;12 | #1;14…#1;15> <56,564> 5 2 (5,5) (161111) (5,5,5),(5,5,5,25),(5,25)4 (5,5,5,5,5) 2 21510 1 0/3 0/0
<57,#1;16…#1;20> <56,564> 5 2 (5,5) (101111) (5,5,5),(5,5,5,5),(5,25)4 (5,5,5,5) 3 2259 1 0/3 0/0
<57,#1;21…#1;30> <56,564> 5 2 (5,5) (101111) (5,5,5),(5,5,5,5),(5,25)4 (5,5,5,5) 3 2159 1 0/3 0/0
<57,#1;1> <56,557> 5 2 (5,5) (212222) (5,25),(5,5,5,5,5),(5,25)4 (5,5,5,5,5) 2 23510 1 1/4 10/10
<57,#1;2> <56,557> 5 2 (5,5) (202222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 2 23511 1 1/4 6/4
<57,#1;3> <56,557> 5 2 (5,5) (222222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 2 21511 1 1/4 7/5
<57,#1;4…#1;6> <56,557> 5 2 (5,5) (212222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 2 23510 1 1/4 10/6
<57,#1;7> <56,557> 5 2 (5,5) (202222) (5,25),(5,5,5,5),(5,25)4 (5,5,5,5) 3 2359 1 0/3 0/0
<57,#1;8> <56,557> 5 2 (5,5) (202222) (5,25),(5,5,5,5),(5,25)4 (5,5,5,5) 3 2159 1 0/3 0/0
<57,#1;1> <56,558> 5 2 (5,5) (212222) (5,25),(5,5,5,5,5),(5,25)4 (5,5,5,5,5) 2 23510 1 1/4 10/10
<57,#1;2> <56,558> 5 2 (5,5) (202222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 2 23511 1 1/4 6/4
<57,#1;3> <56,558> 5 2 (5,5) (222222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 2 21511 1 1/4 7/5
<57,#1;4> <56,558> 5 2 (5,5) (212222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 2 23510 1 1/4 10/6
<57,#1;5…#1;6> <56,558> 5 2 (5,5) (212222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 2 23510 1 1/4 10/6
<57,#1;7> <56,558> 5 2 (5,5) (202222) (5,25),(5,5,5,5),(5,25)4 (5,5,5,5) 3 2359 1 0/3 0/0
<57,#1;8> <56,558> 5 2 (5,5) (202222) (5,25),(5,5,5,5),(5,25)4 (5,5,5,5) 3 2159 1 0/3 0/0
<57,#1;1…#1;3> <56,674> 5 2 (5) (213546) (5,25)2,(5,5,5),(5,25)2,(5,5,5) (5,5,25) 3 2258 1 0/3 0/0
<57,#1;1…#1;2> <56,680> 5 2 (5) (215634) (5,25)6 (5,5,25) 3 2258 1 0/3 0/0
|G| = 390625
<58,#2;1> <56,564> 5 3 (5,5,5) (111111) (5,5,5),(5,5,5,5,5),(5,25)4 (5,5,5,5,5) 3 22511 1 2/4 15/15;32/32
<58,#2;2…#2;5> <56,564> 5 3 (5,5,5) (111111) (5,5,5),(5,5,5,5,5),(5,25)4 (5,5,5,5,5) 3 22511 1 1/3 7/7
<58,#2;6…#2;15> <56,564> 5 3 (5,5,5) (131111) (5,5,5),(5,5,5,5,5),(5,25)4 (5,5,5,5,5) 3 21511 1 1/3 3/3
<58,#2;16> <56,564> 5 3 (5,5,5) (101111) (5,5,5),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 22511 1 2/4 15/15;32/32
<58,#2;17…#2;20> <56,564> 5 3 (5,5,5) (101111) (5,5,5),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 22511 1 1/3 7/7
<58,#2;21…#2;30> <56,564> 5 3 (5,5,5) (121111) (5,5,5),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 21511 1 1/3 3/3
<58,#2;31 | #2;46 | #2;61> <56,564> 5 3 (5,5,5) (111111) (5,5,5),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 22511 1 2/4 15/15;32/32
<58,#2;32…#2;35 | #2;47…#2;50 | #2;62…#2;65> <56,564> 5 3 (5,5,5) (111111) (5,5,5),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 22511 1 1/3 7/7
<58,#2;36…#2;45 | #2;51…#2;60 | #2;66…#2;75> <56,564> 5 3 (5,5,5) (131111) (5,5,5),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 21511 1 1/3 3/3
<58,#2;1> <56,557> 5 3 (5,5,5) (212222) (5,25),(5,5,5,5,5),(5,25)4 (5,5,5,5,5) 3 23511 1 2/4 12/12;26/26
<58,#2;2> <56,557> 5 3 (5,5,5) (212222) (5,25),(5,5,5,5,5),(5,25)4 (5,5,5,5,5) 3 21511 1 2/4 28/28;63/63
<58,#2;3> <56,557> 5 3 (5,5,5) (202222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 23511 1 1/3 6/6
<58,#2;4> <56,557> 5 3 (5,5,5) (222222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 21511 1 1/3 13/13
<58,#2;5 | #2;7 | #2;9> <56,557> 5 3 (5,5,5) (212222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 23511 1 1/3 6/6
<58,#2;6 | #2;8 | #2;10> <56,557> 5 3 (5,5,5) (262222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 21511 1 1/3 13/13
<58,#2;1> <56,558> 5 3 (5,5,5) (212222) (5,25),(5,5,5,5,5),(5,25)4 (5,5,5,5,5) 3 23511 1 2/4 12/12;26/26
<58,#2;2> <56,558> 5 3 (5,5,5) (212222) (5,25),(5,5,5,5,5),(5,25)4 (5,5,5,5,5) 3 21511 1 2/4 28/28;63/63
<58,#2;3> <56,558> 5 3 (5,5,5) (202222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 23511 1 1/3 6/6
<58,#2;4> <56,558> 5 3 (5,5,5) (222222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 21511 1 1/3 13/13
<58,#2;5 | #2;7 | #2;9> <56,558> 5 3 (5,5,5) (212222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 23511 1 1/3 6/6
<58,#2;6 | #2;8 | #2;10> <56,558> 5 3 (5,5,5) (262222) (5,25),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 21511 1 1/3 13/13
<58,#2;1> <56,674> 5 3 (5,5) (213546) a cover of <56,674> (5,25)2,(5,5,5),(5,25)2,(5,5,5) (5,5,25) 3 2259 1 0/2 0/0
<58,#2;2> <56,674> 5 3 (5,5) (213546) a cover of <56,674> (5,25)2,(5,5,5),(5,25)2,(5,5,5) (5,5,25) 3 2259 1 0/2 0/0
<58,#2;3> <56,674> 5 3 (5,5) (213546) (5,25)2,(5,5,5),(5,25)2,(5,5,5) (5,5,25) 3 2359 1 1/3 6/6
<58,#2;1> <56,680> 5 3 (5,5) (215634) a cover of <56,680> (5,25)6 (5,5,25) 3 213159 1 0/2 0/0
<58,#2;2> <56,680> 5 3 (5,5) (215634) (5,25)6 (5,5,25) 3 223159 1 1/3 5/5
<58,#2;3> <56,680> 5 3 (5,5) (215634) a cover of <56,680> (5,25)6 (5,5,25) 3 213159 1 0/2 0/0
|G| = 1953125
<59,#1;6> <56,564>-#2;16 6 3 (5,5,5) (101111) (5,5,5),(5,5,25,25),(5,25)4 (5,5,5,5,25) 3 22513 1 2/5 28/4;65/13
<59,#1;10> <56,564>-#2;16 6 3 (5,5) (101111) (5,5,5),(5,5,5,25),(5,25)4 (5,5,5,5,5) 3 21512 1 2/5 30/3;65/13
|G| = 9765625
<510,#1;11> <56,564>-#2;16-#1;6 7 3 (5,5,5) (101111) (5,5,5),(5,25,25,25),(5,25)4 (5,5,5,25,25) 3 22515 1 1/5 20/15
|G| = 48828125
<511,#2;39> <56,564>-#2;16-#1;6 7 4 (5,5,25) (101111) (5,5,5),(5,25,25,25),(5,25)4 (5,5,5,25,25) 3 22515 1 1/4 15/15
|G| = 244140625
<512,#1;6> <56,564>-#2;16-#1;6-#2;39 8 4 (5,5,25) (101111) (5,5,5),(25,25,25,25),(5,25)4 (5,5,25,25,25) 3 22517 1 2/5 28/6;65/15
<512,#1;12> <56,564>-#2;16-#1;6-#2;39 8 4 (5,5,5) (101111) (5,5,5),(5,25,25,25),(5,25)4 (5,5,5,25,25) 3 22516 1 2/5 16/2;33/7
|G| = 1220703125
<513,#1;1> <56,564>-#2;16-#1;6-#2;39-#1;6 9 4 (5,5,25) (101111) (5,5,5),(25,25,25,125),(5,25)4 (5,25,25,25,25) 3 22519 1 1/5 18/3
|G| = 6103515625
<514,#2;1> <56,564>-#2;16-#1;6-#2;39-#1;6 9 5 (5,25,25) (101111) (5,5,5),(25,25,25,125),(5,25)4 (5,25,25,25,25) 3 22519 1 1/4 13/13


3. Comparison between G(3,2) and G(5,2):


3.1. Similarities:


The top vertices of both coclass graphs can be classified into three kinds.
(The prime p denotes either 3 or 5.)
  1. Roots of infinite descendant trees within G(p,2).

  2. Exactly 2 roots of finite descendant trees within G(p,2),
    <243,4> and <243,9> for p = 3,
    <3125,7> and <3125,10> for p = 5.

  3. Terminal and thus isolated vertices within G(p,2),
    which turn out to be Schur σ-groups, without exceptions.



3.2. Differences:


  1. G(5,2) contains 6 metabelian Schur σ-groups,
    <3125,8>, <3125,9>, <3125,11>, <3125,12>, <3125,13> and <3125,14>,
    whereas G(3,2) contains only 2 metabelian Schur σ-groups,
    <243,5> and <243,7>.

  2. Both finite descendant trees of G(5,2) with roots <15625,674> and <15625,680>
    give rise to non-metabelian Schur σ-groups of order p8 in G(p,3).
    This is the case for only one finite descendant tree of G(3,2) with root <729,45>, but not for <729,57>.

  3. Among the top vertices of G(5,2) there occur 4 roots of infinite descendant trees,
    <3125,3>, <3125,4>, <3125,5> and <3125,6>,
    whereas among the top vertices of G(3,2) there occur only 3 roots of infinite descendant trees,
    <243,3>, <243,6> and <243,8>.

  4. Whereas the infinite descendant trees of G(3,2) with roots <243,6> and <243,8>
    give rise to non-metabelian Schur σ-groups of order p8 in G(p,3),
    this is not the case for the infinite descendant trees of G(5,2)
    with roots <3125,4>, <3125,5> and <3125,6>.

  5. There are no analogues of Ascione's non-CF groups H and I in G(5,2).


*


Karl-Franzens University Graz, right side
Daniel C. Mayer
History:

  • The first diagrams of coclass trees
    were drawn in 1977 by
    J. A. Ascione, G. Havas, and C. R. Leedham-Green .
    They concerned two-generated 3-groups
    of second maximal class, that is, of coclass 2.
    Except for finitely many sporadic groups,
    these groups form 15 coclass trees of the
    coclass graph G(3,2), 7 trees of groups
    with abelianization (3,3) and 8 trees of
    groups with abelianization (3,9).
    The remaining single tree of groups with
    abelianization (3,3,3) was skipped, since
    the authors considered two-generated groups.
    The periodic structure of the branches of
    5 trees (B,Q,U and A,G) with metabelian mainlines
    could be recognized at this early stage.

  • In her 1979 Thesis, J. A. Ascione
    extended the diagrams to groups of bigger order
    and included 2- and 3-groups of maximal class.
    She also gave the first detailed description
    of the p-group generation algorithm for p = 3
    and two-generated 3-groups
    with solvable automorphism group.
    This algorithm was implemented in full generality
    by E. A. O'Brien in 1990 and was named after him.
    However, the proof for the periodicity of branches
    was given later by M. du Sautoy in 2001 and,
    independently with different methods,
    by B. Eick and C. R. Leedham-Green in 2008.

  • In 1989, B. Nebelung presented
    the coclass trees of all metabelian
    3-groups with abelianization (3,3).
    In order to reduce the problem to
    a collection of finitely many diagrams,
    she had to draw the single tree of G(3,1),
    three trees of G(3,2), and then to use
    the periodicity of the structure of
    metabelian coclass trees of G(3,r), r ≥ 3,
    which contains four trees for odd r
    and six trees for even r.
    A theoretical proof for this periodicity of
    coclass trees in the metabelian skeleton
    of coclass graphs is unknown till now.

*
Web master's e-mail address:
contact@algebra.at
*

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