# Properties

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

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## 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 Nilpotency class: $3$ 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

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

## Low degree siblings

8T15, 16T35, 16T38 x 2, 16T45, 32T21

Siblings are shown with degree $\leq 47$

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