# Properties

 Label 8T39 Order $192$ n $8$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_2^3:S_4$

# Related objects

## Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $39$
Group :  $C_2^3:S_4$
CHM label :  $[2^{3}]S(4)$
Parity:  $1$
Primitive:  No
Generators:   (1,2,3,7)(4,8,6,5), (1,6)(3,8,4,7)
$|\textrm{Aut}(F/K)|$:  $2$
Low degree resolvents:
 2: 2T1 6: 3T2 24: 4T5, 4T5, 4T5 96: 8T34

## Subfields

Degree 2: None

Degree 4: $S_4$

## Low degree siblings

8T39b, 8T39c, 8T39d, 8T39e, 8T39f, 16T442a, 16T442b, 16T442c
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1$ $6$ $2$ $(3,4)(7,8)$ $2, 2, 1, 1, 1, 1$ $12$ $2$ $(3,7)(4,8)$ $4, 2, 1, 1$ $24$ $4$ $(2,3,5,4)(7,8)$ $3, 3, 1, 1$ $32$ $3$ $(2,3,7)(4,8,5)$ $2, 2, 2, 2$ $12$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $2, 2, 2, 2$ $6$ $2$ $(1,2)(3,7)(4,8)(5,6)$ $2, 2, 2, 2$ $6$ $2$ $(1,2)(3,8)(4,7)(5,6)$ $6, 2$ $32$ $6$ $(1,2,3,6,5,4)(7,8)$ $4, 4$ $24$ $4$ $(1,2,3,7)(4,8,6,5)$ $4, 4$ $24$ $4$ $(1,2,3,8)(4,7,6,5)$ $4, 4$ $12$ $4$ $(1,2,6,5)(3,7,4,8)$ $2, 2, 2, 2$ $1$ $2$ $(1,6)(2,5)(3,4)(7,8)$

## Group invariants

 Order: $192=2^{6} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes
 Character table: ``` 2 6 5 4 3 1 4 5 5 1 3 3 4 6 3 1 . . . 1 . . . 1 . . . 1 1a 2a 2b 4a 3a 2c 2d 2e 6a 4b 4c 4d 2f 2P 1a 1a 1a 2a 3a 1a 1a 1a 3a 2d 2e 2f 1a 3P 1a 2a 2b 4a 1a 2c 2d 2e 2f 4b 4c 4d 2f 5P 1a 2a 2b 4a 3a 2c 2d 2e 6a 4b 4c 4d 2f X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 -1 1 1 1 -1 -1 1 1 X.3 2 2 . . -1 . 2 2 -1 . . 2 2 X.4 3 -1 -1 1 . -1 3 -1 . -1 1 -1 3 X.5 3 -1 1 -1 . 1 3 -1 . 1 -1 -1 3 X.6 3 3 -1 -1 . -1 -1 -1 . 1 1 -1 3 X.7 3 3 1 1 . 1 -1 -1 . -1 -1 -1 3 X.8 3 -1 -1 1 . -1 -1 3 . 1 -1 -1 3 X.9 3 -1 1 -1 . 1 -1 3 . -1 1 -1 3 X.10 4 . 2 . 1 -2 . . -1 . . . -4 X.11 4 . -2 . 1 2 . . -1 . . . -4 X.12 6 -2 . . . . -2 -2 . . . 2 6 X.13 8 . . . -1 . . . 1 . . . -8 ```