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

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
Generators:   (1,6,2,4)(3,5,8,7), (1,6,8,4)(2,5,3,7)
$|\Aut(F/K)|$:  $1$
Low degree resolvents:  
2: 2T1
6: 3T2
24: 4T5, 4T5, 4T5

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

12T66a, 12T66b, 12T66c, 12T67, 12T68a, 12T68b, 12T68c, 12T69, 16T194
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  .  .  .  .