# Properties

 Label 8T32 Order $$96$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $((C_2 \times D_4): C_2):C_3$

# Related objects

## Group action invariants

 Degree $n$ : $8$ Transitive number $t$ : $32$ Group : $((C_2 \times D_4): C_2):C_3$ CHM label : $[2^{3}]A(4)$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,3)(2,8)(4,6)(5,7), (2,5)(3,4), (1,8)(2,3)(4,5)(6,7), (1,2,3)(4,6,5), (1,5)(2,6)(3,7)(4,8) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
12:  $A_4$ x 5
48:  $C_2^4:C_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 4: $A_4$

## Low degree siblings

8T32 x 2, 24T97 x 3, 24T149, 32T420

Siblings are shown with degree $\leq 47$

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