 # 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 2nd 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 1st 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) G(4,5) 1*D.5, 1*D.10 Unique State G(4,5) 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) 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(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) G(6,9) 3*F.7, 2*F.11, 4*F.12, 4*F.13 Ground State G(8,9) 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) 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 2nd 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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