# 2-Stage Metabelian 3-Groups

## 3.A. All known sextic S3 examples:

In the following diagrams, we give all 2-stage metabelian 3-groups G = G(K*1|k0) that occured for sextic S3 fields K over the quadratic cyclotomic field k0 = Q((-3)1/2) having 2-prime conductors f = 32*q with a prime q = -1(mod 9) [3]. We denote by G = G1 > G2 > ... > Gi > ... > Gm = 1 the descending central series of class m - 1 of G with Gi+1 = [Gi,G]. In particular, G2 = [G,G] is the commutator subgroup G' of G.
The relative 3-genus field K* = (K|k0)* of K over k0 is the compositum K* = k*k~ of k = k0(q1/3) and k~ = k0(31/3), the cubic Kummer extensions of k0 with the two primeconductors q and 9 dividing f. Hence K* is a bicyclic bicubic relative extension of k0 that contains yet two other cubic Kummer extensions K = k0((3q)1/3) and K~ = k0((9q)1/3) of k0 with conductor f.

We gratefully acknowledge that Karim Belabas has computed the structure of G2 = Syl3C(K*) with the aid of PARI.
Group of maximal class
(e = 2)
G = G(5)(alpha,beta,gamma)
in ZEF 2a(5,5)
with (alpha,beta,gamma) = (0,0,0),
of class 4 and of order 35 = 243
 K*1 = K~1 | F4 | K1 | K* / / \ \ k K K~ k~ \ \ / / k0
 G5 = 1 | G4 = (3) | G3 = (3,3) | G2 = (32,3) / / \ \ h H H~ h~ \ \ / / G
Minimal occurrence:
f = 153 = 32*17
3-class numbers of K and K~:
(9,81)
principal factorization types of K and K~:
(BETA,GAMMA) [3].
Group of maximal class
(e = 2)
G = G(4)(alpha,beta,gamma)
in ZEF 2a(4,4)
with (alpha,beta,gamma) = (1,1,0),
of class 3 and of order 34 = 81
 K*1 | K1 = K~1 | K* / / \ \ k K K~ k~ \ \ / / k0
 G4 = 1 | G3 = (3) | G2 = (3,3) / / \ \ h H H~ h~ \ \ / / G
Minimal occurrence:
f = 801 = 32*89
3-class numbers of K and K~:
(9,9)
principal factorization types of K and K~:
(BETA,BETA) [3].
Group of lower than second maximal class
(e = 4)
G = G(6,8)((alpha,beta,gamma,delta),rho)
in ZEF(6,8)
of class 5 and of order 38 = 6561
 K*1 | F5 || K1 = K~1 || F3 | K* / / \ \ k K K~ k~ \ \ / / k0
 G6 = 1 | G5 = (3) || G4 = (3,3,3) || G3 = (32,3,3,3) | G2 = (32,32,3,3) / / \ \ h H H~ h~ \ \ / / G
Minimal occurrence:
f = 2421 = 32*269
3-class numbers of K and K~:
(81,81)
principal factorization types of K and K~:
(BETA,BETA) [3].
Group of lower than second maximal class
(e = 4)
G = G(7,9)((alpha,beta,gamma,delta),rho)
in ZEF2(7,9)
of class 6 and of order 39 = 19683
 K*1 | F6 | K~1 || K1 || F3 | K* / / \ \ k K K~ k~ \ \ / / k0
 G7 = 1 | G6 = (3) | G5 = (3,3) || G4 = (32,3,3) || G3 = (32,32,3,3) | G2 = (33,32,3,3) / / \ \ h H H~ h~ \ \ / / G
Minimal occurrence:
f = 7443 = 32*827
3-class numbers of K and K~:
(81,729)
principal factorization types of K and K~:
(BETA,GAMMA) [3].
Group of lower than second maximal class
(e = 4)
G = G(9,11)((alpha,beta,gamma,delta),rho)
in ZEF2(9,11)
of class 8 and of order 311 = 177147
 K*1 | F8 | K~1 | F6 | F5 || K1 || F3 | K* / / \ \ k K K~ k~ \ \ / / k0
 G9 = 1 | G8 = (3) | G7 = (3,3) | G6 = (32,3) | G5 = (32,32) || G4 = (33,32,3) || G3 = (33,33,3,3) | G2 = (34,33,3,3) / / \ \ h H H~ h~ \ \ / / G
Minimal occurrence:
f = 9873 = 32*1097
3-class numbers of K and K~:
(81,6561)
principal factorization types of K and K~:
(BETA,GAMMA) [3].
 References: [1] Brigitte Nebelung, Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Köln, 1989 [2] Aïssa Derhem, Retour sur la thèse de Moulay Chrif Ismaïli, Casablanca, 2003 [3] Daniel C. Mayer, Class Numbers and Principal Factorization Types of Multiplets of Pure Cubic Fields Q( R1/3 ) with R < 106, Univ. Graz, Computer Centre, 2003

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