# Properties

 Label 8T20 Order $$32$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_2^3: C_4$

# Related objects

## Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $20$
Group :  $C_2^3: C_4$
CHM label :  $[2^{3}]4$
Parity:  $1$
Primitive:  No
Generators:   (1,2,3,8)(4,5,6,7), (2,6)(3,7)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:
 2: 2T1, 2T1, 2T1 4: 4T1, 4T1, 4T2 8: 4T3, 4T3, 8T2 16: 8T10

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

## Low degree siblings

8T19a, 8T19b, 8T21, 16T33a, 16T33b, 16T52, 16T53
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

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

## Group invariants

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