# Properties

 Label 8T32 Order $$96$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No

# Learn more about

## Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $32$
CHM label :  $[2^{3}]A(4)$
Parity:  $1$
Primitive:  No
Generators:   (2,8,3)(4,5,7), (1,4,8)(3,7,6), (1,7,2,6,8,5)(3,4)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:
 3: 3T1 12: 4T4, 4T4, 4T4, 4T4, 4T4 48: 12T32

## Subfields

Degree 2: None

Degree 4: $A_4$

## Low degree siblings

8T32b, 8T32c
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1$ $6$ $2$ $(3,4)(7,8)$ $3, 3, 1, 1$ $16$ $3$ $(2,3,7)(4,8,5)$ $3, 3, 1, 1$ $16$ $3$ $(2,7,3)(4,5,8)$ $2, 2, 2, 2$ $6$ $2$ $(1,2)(3,7)(4,8)(5,6)$ $2, 2, 2, 2$ $6$ $2$ $(1,2)(3,8)(4,7)(5,6)$ $6, 2$ $16$ $6$ $(1,2,3,6,5,4)(7,8)$ $4, 4$ $6$ $4$ $(1,2,6,5)(3,7,4,8)$ $4, 4$ $6$ $4$ $(1,2,6,5)(3,8,4,7)$ $6, 2$ $16$ $6$ $(1,2,7,6,5,8)(3,4)$ $2, 2, 2, 2$ $1$ $2$ $(1,6)(2,5)(3,4)(7,8)$

## Group invariants

 Order: $96=2^{5} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes GAP id: [96, 204]
 Character table:  2 5 4 1 1 4 4 1 4 4 1 5 3 1 . 1 1 . . 1 . . 1 1 1a 2a 3a 3b 2b 2c 6a 4a 4b 6b 2d 2P 1a 1a 3b 3a 1a 1a 3a 2d 2d 3b 1a 3P 1a 2a 1a 1a 2b 2c 2d 4a 4b 2d 2d 5P 1a 2a 3b 3a 2b 2c 6b 4a 4b 6a 2d X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 A /A 1 1 /A 1 1 A 1 X.3 1 1 /A A 1 1 A 1 1 /A 1 X.4 3 3 . . -1 -1 . -1 -1 . 3 X.5 3 -1 . . 3 -1 . -1 -1 . 3 X.6 3 -1 . . -1 3 . -1 -1 . 3 X.7 3 -1 . . -1 -1 . -1 3 . 3 X.8 3 -1 . . -1 -1 . 3 -1 . 3 X.9 4 . 1 1 . . -1 . . -1 -4 X.10 4 . A /A . . -/A . . -A -4 X.11 4 . /A A . . -A . . -/A -4 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3