Tree Diagram of Metabelian 3-Groups G

with Commutator Factor Group G/G´ of Type (3,3)

The tree consists of the root G⁽²⁾ ≅ (3,3),
the infinite trunk G⁽³⁾ = G^(3,3),G^(4,5),G^(5,7),…,G^(m,2m-3),… ,
up to six infinite branches starting with each node on the trunk,
finite twigs, and leaves (terminal nodes).

We restrict the tree to (isomorphism classes of) groups G in CBF(m,n) of order 3ⁿ and class m-1
which occur as automorphism groups Gal(K₂|K) for quadratic base fields K with discriminant between -10⁶ and 10⁷.
K₂ denotes the 2^nd Hilbert 3-class field of K.

The leaves (and certain other nodes¹⁾) are represented together with their multiplet of transfer types
distinguishing ground states and excited states. (These concepts are dependent on the kind of base field!)

The finite twigs with the leaves of transfer types D.5, D.10, and the ground states of G.19, H.4
are represented in a position above the groups of lower than second maximal class only to save space.
Their nodes are groups G of second maximal class and G´ is of rank 3, with the exception of G.19.

¹⁾ Since the 1^st of January 2008 we know that an internal node can occur as an automorphism group Gal(K₂|K),
as the transfer type c.21 shows.

n=2

G⁽²⁾

n=3

G⁽³⁾

n=4

G⁽⁴⁾

G⁽⁴⁾
1*a.2,
1*a.3
1*a.3*
Ground
State

n=5

G^(4,5)

G^(4,5)

G^(4,5)
1*D.5,
1*D.10
Unique
State

G^(4,5)

G⁽⁵⁾

n=6

G^(5,6)
2*G.19
Ground
State

G^(5,6)
4*H.4
Ground
State

G^(5,6)
1*c.18
Ground
State

G^(5,6)
1*c.21
Ground
State

G⁽⁶⁾
3*a.1
Ground
State

G⁽⁶⁾

G⁽⁶⁾
1*a.2,
2*a.3
Excited
State

n=7

G^(5,7)

G^(6,7)

G^(6,7)
1*E.6,
2*E.14
Ground
State

G^(6,7)
1*E.8,
2*E.9
Ground
State

G^(6,7)

G⁽⁷⁾

n=8

G^(6,8)
6*b.10
Ground
State

G^(6,8)
2*d.19
1*d.23
2*d.25
Ground
State

G^(7,8)
8*H.4
Excited
State

G^(7,8)
1*c.18
Excited
State

G^(7,8)
1*c.21
Excited
State

G^(7,8)
8*G.16
Ground
State

G⁽⁸⁾
3*a.1
Excited
State

G⁽⁸⁾

n=9

G^(6,9)

G^(6,9)

G^(6,9)
3*F.7,
2*F.11,
4*F.12,
4*F.13
Ground
State

G^(8,9)

G^(8,9)
1*E.6,
2*E.14
Excited
State

G^(8,9)
1*E.8,
2*E.9
Excited
State

G^(8,9)

...

n=10

G^(7,10)
2*d.19*
1*d.23*
2*d.25*
Ground
State

G^(7,10)
7*G.16,
10*G.19,
13*H.4
Variant

G^(9,10)
8*H.4
Second
Excited
State

...

G^(9,10)
8*G.16
Excited
State

n=11

G^(7,11)

G^(8,11)
4*F.7,
4*F.11,
8*F.12,
8*F.13
Excited
State

G^(8,11)

n=12

G^(9,12)
10*G.16,
16*G.19,
20*H.4
Variant
Excited
State

n=13

G^(8,13)

G^(8,13)
3*F.7,
2*F.11,
4*F.12,
4*F.13
Variant

...

G´ of

Rank 4

Rank 3

Rank 2

Class

lower

2^nd maximal

maximal

References:

[Ne] Brigitte Nebelung,
Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem,
Inauguraldissertation, Univ. zu Köln, 1989.

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