3.A. All known sextic S_{3} examples:
In the following diagrams,
we give all 2stage metabelian 3groups
G = G(K*_{1}k_{0})
that occured for sextic S_{3} fields K
over the quadratic cyclotomic field k_{0} = Q((3)^{1/2})
having 2prime conductors f = 3^{2}*q with a prime q = 1(mod 9)
[3].
We denote by
G = G_{1} > G_{2} > ... > G_{i} > ... > G_{m} = 1
the descending central series of class m  1
of G with G_{i+1} = [G_{i},G].
In particular, G_{2} = [G,G] is the commutator subgroup G' of G.
The relative 3genus field K* = (Kk_{0})* of K over k_{0} is the compositum
K* = k*k~ of k = k_{0}(q^{1/3}) and k~ = k_{0}(3^{1/3}),
the cubic Kummer extensions of k_{0} with the two primeconductors q and 9 dividing f.
Hence K* is a bicyclic bicubic relative extension of k_{0}
that contains yet two other cubic Kummer extensions
K = k_{0}((3q)^{1/3}) and K~ = k_{0}((9q)^{1/3})
of k_{0} with conductor f.
We gratefully acknowledge that Karim Belabas has computed
the structure of G_{2} = Syl_{3}C(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 3^{5} = 243

    K*_{1} = K~_{1}     
         
    F_{4}     
         
    K_{1}     
         
    K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    k_{0}     

    G_{5} = 1     
         
    G_{4} = (3)     
         
    G_{3} = (3,3)     
         
    G_{2} = (3^{2},3)     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 153 = 3^{2}*17
3class 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 3^{4} = 81

    K*_{1}     
         
    K_{1} = K~_{1}     
         
    K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    k_{0}     

    G_{4} = 1     
         
    G_{3} = (3)     
         
    G_{2} = (3,3)     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 801 = 3^{2}*89
3class 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 3^{8} = 6561

    K*_{1}     
         
    F_{5}     
         
    K_{1} = K~_{1}     
         
    F_{3}     
         
    K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    k_{0}     

    G_{6} = 1     
         
    G_{5} = (3)     
         
    G_{4} = (3,3,3)     
         
    G_{3} = (3^{2},3,3,3)     
         
    G_{2} = (3^{2},3^{2},3,3)     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 2421 = 3^{2}*269
3class 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 3^{9} = 19683

    K*_{1}     
         
    F_{6}     
         
    K~_{1}     
         
    K_{1}     
         
    F_{3}     
         
    K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    k_{0}     

    G_{7} = 1     
         
    G_{6} = (3)     
         
    G_{5} = (3,3)     
         
    G_{4} = (3^{2},3,3)     
         
    G_{3} = (3^{2},3^{2},3,3)     
         
    G_{2} = (3^{3},3^{2},3,3)     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 7443 = 3^{2}*827
3class 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 3^{11} = 177147

    K*_{1}     
         
    F_{8}     
         
    K~_{1}     
         
    F_{6}     
         
    F_{5}     
         
    K_{1}     
         
    F_{3}     
         
    K*     
 /   /   \   \  
k   K     K~   k~ 
 \   \   /   /  
    k_{0}     

    G_{9} = 1     
         
    G_{8} = (3)     
         
    G_{7} = (3,3)     
         
    G_{6} = (3^{2},3)     
         
    G_{5} = (3^{2},3^{2})     
         
    G_{4} = (3^{3},3^{2},3)     
         
    G_{3} = (3^{3},3^{3},3,3)     
         
    G_{2} = (3^{4},3^{3},3,3)     
 /   /   \   \  
h   H     H~   h~ 
 \   \   /   /  
    G     

Minimal occurrence:
f = 9873 = 3^{2}*1097
3class numbers of K and K~:
(81,6561)
principal factorization types of K and K~:
(BETA,GAMMA)
[3].


References:
[1] Brigitte Nebelung,
Klassifikation metabelscher 3Gruppen
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( R^{1/3} ) with R < 10^{6},
Univ. Graz, Computer Centre, 2003

