# Properties

 Label 8T22 Order $32$ n $8$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_2^3 : D_4$

# Related objects

## Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $22$
Group :  $C_2^3 : D_4$
CHM label :  $E(8):D_{4}=[2^{3}]2^{2}$
Parity:  $1$
Primitive:  No
Generators:   (2,3)(6,7), (2,3)(4,5), (1,3)(2,8)(4,6)(5,7), (1,5)(2,6)(3,7)(4,8), (1,8)(2,3)(4,5)(6,7)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:
 2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1 4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2 8: 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3 16: 16T3

## Subfields

Degree 2: $C_2$, $C_2$, $C_2$

Degree 4: $V_4$

## Low degree siblings

8T22b, 8T22c, 8T22d, 8T22e, 8T22f, 16T23a, 16T23b, 16T23c, 16T23d, 16T23e, 16T23f, 16T23g, 16T23h, 16T23i
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 49]
 Character table: ``` 2 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 1a 2a 2b 2c 2d 2e 4a 4b 2f 4c 2g 4d 2h 4e 4f 2i 2j 2P 1a 1a 1a 1a 1a 1a 2j 2j 1a 2j 1a 2j 1a 2j 2j 1a 1a 3P 1a 2a 2b 2c 2d 2e 4a 4b 2f 4c 2g 4d 2h 4e 4f 2i 2j X.1 1 1 1 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 -1 -1 1 1 X.3 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 X.4 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 X.5 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 X.6 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 X.7 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 X.8 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 X.9 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 X.10 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 X.11 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 X.12 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 X.13 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 X.14 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 X.15 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 X.16 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 X.17 4 . . . . . . . . . . . . . . . -4 ```