# 2-Stage Metabelian 3-Groups

## 2.A. All known cyclic cubic examples:

In the following diagrams, we give all 2-stage metabelian 3-groups G = G(K*1|Q) that occured for cyclic cubic fields K with 2-prime conductors f < 105 [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 absolute 3-genus field K* is the compositum K* = k*k~ of k and k~, the cyclic cubic fields with the two primeconductors dividing f. Hence K* is a bicyclic bicubic field that contains yet another cyclic cubic field K~ with conductor f.
Group of maximal class
(e = 2)
G = (3,3)
in ZEF(2,2)
of class 1 and of order 32 = 9
 K*1 = K* / / \ \ k K K~ k~ \ \ / / Q
 G2 = 1 / / \ \ h H H~ h~ \ \ / / G
Minimal occurrence:
f = 63 = 32*7
3-class numbers of K and K~:
(3,3)
Group of maximal class
(e = 2)
G = G(3)(alpha,beta,gamma)
in ZEF 1a(3,3)
with (alpha,beta,gamma) = (0,0,0),
of class 2 and of order 33 = 27
 K*1 | K* / / \ \ k K K~ k~ \ \ / / Q
 G3 = 1 | G2 = (3) / / \ \ h H H~ h~ \ \ / / G
Minimal occurrence:
f = 657 = 32*73
3-class numbers of K and K~:
(9,9)
Group of maximal class
(e = 2)
G = G(4)(alpha,beta,gamma)
in ZEF 2a(4,4)
with (alpha,beta,gamma) = (0,1,0),
of class 3 and of order 34 = 81
 K*1 | F3 | K* / / \ \ k K K~ k~ \ \ / / Q
 G4 = 1 | G3 = (3) | G2 = (3,3) / / \ \ h H H~ h~ \ \ / / G
Minimal occurrence:
f = 2439 = 32*271
3-class numbers of K and K~:
(9,9)
Group of second maximal class
(e = 3)
G = G(4,5)((alpha,beta,gamma,delta),rho)
in ZEF 1a(4,5)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = 0,
of class 3 and of order 35 = 243
 K*1 || F3 | K* / / \ \ k K K~ k~ \ \ / / Q
 G4 = 1 || G3 = (3,3) | G2 = (3,3,3) / / \ \ h H H~ h~ \ \ / / G
Minimal occurrence:
f = 4711 ("Eau de Cologne") = 7*673
3-class numbers of K and K~:
(27,27)
Group of second maximal class
(e = 3)
G = G(5,6)((alpha,beta,gamma,delta),rho)
in ZEF 1b(5,6) resp. ZEF 2a(5,6)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = 1
resp. rho = 0,
of class 4 and of order 36 = 729
with rho != beta - 1 resp. rho = 0
 K*1 | F4 || F3 | K* / / \ \ k K K~ k~ \ \ / / Q
 G5 = 1 | G4 = (3) || G3 = (3,3,3) | G2 = (32,3,3) / / \ \ h H H~ h~ \ \ / / G
Minimal occurrence:
f = 5383 = 7*769 resp.
f = 41977 = 13*3229
3-class numbers of K and K~:
(27,27) resp. (81,27)
Group of second maximal class
(e = 3)
G = G(5,6)((alpha,beta,gamma,delta),rho)
in ZEF 1b(5,6)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = -1,
of class 4 and of order 36 = 729
with rho = beta - 1 != 0
 K*1 | F4 || F3 | K* / / \ \ k K K~ k~ \ \ / / Q
 G5 = 1 | G4 = (3) || G3 = (3,3,3) | G2 = (3,3,3,3) / / \ \ h H H~ h~ \ \ / / G
Minimal occurrence:
f = 21763 = 7*3109
3-class numbers of K and K~:
(27,27)

Remark:
Here, the difference to f = 5383 is that we have G2 = (3,3,3,3) and rho = -1 instead of G2 = (32,3,3) and rho = +1, which implies different structures of Syl3C(K*) = G(K*1|K*) = G2.
Group of second maximal class
(e = 3)
G = G(6,7)((alpha,beta,gamma,delta),rho)
in ZEF 2b(6,7)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = +-1,
of class 5 and of order 37 = 2187
 K*1 | F5 | F4 || F3 | K* / / \ \ k K K~ k~ \ \ / / Q
 G6 = 1 | G5 = (3) | G4 = (3,3) || G3 = (32,3,3) | G2 = (32,32,3) / / \ \ h H H~ h~ \ \ / / G
Minimal occurrence:
f = 68857 = 37*1861
3-class numbers of K and K~:
(81,27)
Group of second maximal class
(e = 3)
G = G(7,8)((alpha,beta,gamma,delta),rho)
in ZEF 2b(7,8)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = +-1,
of class 6 and of order 38 = 6561
 K*1 | F6 | F5 | F4 || F3 | K* / / \ \ k K K~ k~ \ \ / / Q
 G7 = 1 | G6 = (3) | G5 = (3,3) | G4 = (32,3) || G3 = (32,32,3) | G2 = (33,32,3) / / \ \ h H H~ h~ \ \ / / G
Minimal occurrence:
f = 36667 = 37*991
3-class numbers of K and K~:
(243,27)
Group of lower than second maximal class
(e = 4)
G = G(6,8)((alpha,beta,gamma,delta),rho)
in ZEF 2a(6,8)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = 0,
of class 5 and of order 38 = 6561
 K*1 | F5 || F4 || F3 | K* / / \ \ k K K~ k~ \ \ / / Q
 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 = 42127 = 103*409
3-class numbers of K and K~:
(243,81)
 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, Sur les corps cubiques cycliques de conducteur divisible par deux premiers, Casablanca, 2002 [3] Daniel C. Mayer, Class Numbers and Principal Factorizations of Families of Cyclic Cubic Fields with Discriminant d < 1010, Univ. Graz, Computer Centre, 2002

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