# Properties

 Label 8T15 Order $$32$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $Z_8 : Z_8^\times$

# Related objects

## Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $15$
Group :  $Z_8 : Z_8^\times$
CHM label :  $[1/4.cD(4)^{2}]2$
Parity:  $-1$
Primitive:  No
Generators:   (1,2,3,4,5,6,7,8), (1,5)(3,7), (1,6)(2,5)(3,4)(7,8)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:
 2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1 4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2 8: 4T3, 4T3, 8T3 16: 8T9

## Subfields

Degree 2: $C_2$

Degree 4: $D_4$

## Low degree siblings

8T15b, 16T35, 16T38a, 16T38b, 16T45
A number field with this Galois group has exactly one arithmetically equivalent field.

## Conjugacy Classes

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 43]
 Character table:  2 5 3 4 3 3 3 3 3 4 4 5 1a 2a 2b 2c 2d 8a 4a 8b 4b 4c 2e 2P 1a 1a 1a 1a 1a 4b 2e 4b 2e 2e 1a 3P 1a 2a 2b 2c 2d 8a 4a 8b 4b 4c 2e 5P 1a 2a 2b 2c 2d 8a 4a 8b 4b 4c 2e 7P 1a 2a 2b 2c 2d 8a 4a 8b 4b 4c 2e X.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 X.3 1 -1 -1 1 1 1 -1 -1 1 -1 1 X.4 1 -1 1 -1 -1 1 -1 1 1 1 1 X.5 1 -1 1 -1 1 -1 1 -1 1 1 1 X.6 1 1 -1 -1 -1 1 1 -1 1 -1 1 X.7 1 1 -1 -1 1 -1 -1 1 1 -1 1 X.8 1 1 1 1 -1 -1 -1 -1 1 1 1 X.9 2 . 2 . . . . . -2 -2 2 X.10 2 . -2 . . . . . -2 2 2 X.11 4 . . . . . . . . . -4