# Properties

 Label 8T19 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 $t$ :  $19$
Group :  $C_2^3 : C_4$
CHM label :  $E(8):4=[1/4.eD(4)^{2}]2$
Parity:  $1$
Primitive:  No
Generators:  (1,4)(2,7)(3,6)(5,8), (1,6,8,5)(2,7,3,4)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:
 2: $C_2$ x 3 4: $C_4$ x 2, $V_4$ 8: $D_4$ x 2, $C_4\times C_2$ 16: $C_2^2:C_4$

## Subfields

Degree 2: $C_2$

Degree 4: $D_4$

## Low degree siblings

8T19, 8T20, 8T21, 16T33 x 2, 16T52, 16T53, 32T19
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

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

## Group invariants

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