Properties

Label 8T34
Order \(96\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $V_4^2:S_3$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $34$
Group :  $V_4^2:S_3$
CHM label :  $1/2[E(4)^{2}:S_{3}]2=E(4)^{2}:D_{6}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8)(2,3), (1,2,3)(5,6,7), (1,5)(2,7)(3,6)(4,8)
$|\Aut(F/K)|$:  $1$

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

Group invariants

Order:  $96=2^{5} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [96, 227]
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  .  .  .  .