Theoretical foundations for metabelian 3-groups G with G/G' of type (9,3)
Whereas the body G' = G2 > G3 > … > Gm-1 > Gm = 1,
that is the lower central series starting with the commutator subgroup G',
of a metabelian 3-group G having abelianization G/G' of type (9,3)
shows many similarities to an isoclinic 3-group G with G/G' of type (3,3),
the head, consisting of all normal subgroups U between G and G',
has an intrinsic polarization induced by the factor group G/G' ≅ (9,3),
in remarkable contrast to the perfect symmetry in the case G/G' ≅ (3,3).
The polarization is caused by the maximal subgroup M4
with bicyclic quotient M4/G'
and by the Frattini subgroup Φ(G) = M˜4 of index 9 in G.
Consequently, the transfer kernel type (TKT) κ(G) must be
replaced by the punctured TKT (κ(1),κ(2),κ(3);κ(4)) which is defined as a
double orbit under the action of S3×S3
instead of a (simple) orbit under S4.
These two actions yield completely different partitions
of the set of all quartets of digits between 0 and 4 into orbits,
a fact which has never been treated adequately up to now.
A further polarization is due to Blackburn's 2-step centralizer χ2(G)
of G2/G4 in G/G4,
which coincides either with M4 (simple polarization) or M1 (bipolarization)
for suitable selection of generators.
Finally, a third polarization is occasionally induced by the power structure of G.
This can lead to the completely unknown phenomenon of a triple polarization,
which cannot occur for G/G' ≅ (3,3)),
and needs a great deal of detailed investigation for groups G of coclass cc(G) ≥ 5.
Computational techniques for metabelian 3-groups G with G/G' of type (9,3):
Top-down principalization algorithm via class group structure for quadratic fields of type (9,3)
Since the copyright has been transferred already to World Scientific Publishing,
the International Journal of Number Theory, Springer, and Monatshefte für Mathematik,
we are now going to grant public access to the PARI/GP source code implementing the algorithm:
The output file FungWilliamsPol93.csv of the first step
must be opened with Microsoft Office / Excel,
ordered by the first column in descending order,
and split into simple text files D10.gp, Q10.gp, L10.gp, A10.gp
containing the first, second, third, fourth column.
These four files are used as input for the second step.
The output file FrattiniRemainderCmpl93.csv of the second step
must be split into files DH10.gp, QH10.gp, LH10.gp, AH10.gp,
which are used as input for the third step.
The output file LlorenteQuerPol93.csv of the first step and
the output file FrattiniRemainderReal93.csv of the second step
must be processed as above, except that there is no file Q10.gp,
since the constructed polynomials are trace free,
and D10.gp, L10.gp, A10.gp are called Dsc10.gp, Lin10.gp, Abs10.gp.
Second 3-class groups G32(K) of number fields K with 3-class group Cl3(K) of type (9,3)
Distribution of the groups G32(K) on the coclass graph G(3,2)
Details for Quadratic Fields with special Punctured Principalization Types