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Properties

Label 8T34
Degree $8$
Order $96$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $V_4^2:S_3$

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Show commands: Magma

magma: G := TransitiveGroup(8, 34);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $34$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $V_4^2:S_3$
CHM label:  $1/2[E(4)^{2}:S_{3}]2=E(4)^{2}:D_{6}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8)(2,3), (1,2,3)(5,6,7), (1,5)(2,7)(3,6)(4,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$24$:  $S_4$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

12T66 x 3, 12T67, 12T68 x 3, 12T69, 16T194, 24T195 x 3, 24T196 x 3, 24T197 x 3, 24T198, 24T199, 24T200 x 3, 32T398

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1 $ $6$ $2$ $(4,5)(6,7)$
$ 3, 3, 1, 1 $ $32$ $3$ $(2,3,8)(5,7,6)$
$ 2, 2, 2, 2 $ $3$ $2$ $(1,2)(3,8)(4,5)(6,7)$
$ 2, 2, 2, 2 $ $3$ $2$ $(1,2)(3,8)(4,6)(5,7)$
$ 2, 2, 2, 2 $ $3$ $2$ $(1,2)(3,8)(4,7)(5,6)$
$ 2, 2, 2, 2 $ $12$ $2$ $(1,4)(2,5)(3,6)(7,8)$
$ 4, 4 $ $12$ $4$ $(1,4,2,5)(3,6,8,7)$
$ 4, 4 $ $12$ $4$ $(1,4,3,6)(2,5,8,7)$
$ 4, 4 $ $12$ $4$ $(1,4,8,7)(2,5,3,6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $96=2^{5} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  96.227
magma: IdentifyGroup(G);
 
Character table:    
      2  5  4  .  5  5  5  3  3  3  3
      3  1  .  1  .  .  .  .  .  .  .

        1a 2a 3a 2b 2c 2d 2e 4a 4b 4c
     2P 1a 1a 3a 1a 1a 1a 1a 2b 2d 2c
     3P 1a 2a 1a 2b 2c 2d 2e 4a 4b 4c

X.1      1  1  1  1  1  1  1  1  1  1
X.2      1  1  1  1  1  1 -1 -1 -1 -1
X.3      2  2 -1  2  2  2  .  .  .  .
X.4      3 -1  . -1  3 -1 -1  1  1 -1
X.5      3 -1  . -1  3 -1  1 -1 -1  1
X.6      3 -1  .  3 -1 -1 -1 -1  1  1
X.7      3 -1  .  3 -1 -1  1  1 -1 -1
X.8      3 -1  . -1 -1  3 -1  1 -1  1
X.9      3 -1  . -1 -1  3  1 -1  1 -1
X.10     6  2  . -2 -2 -2  .  .  .  .

magma: CharacterTable(G);