# Properties

 Label 8T33 Order $$96$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_2^4:C_6$

# Related objects

## Group action invariants

 Degree $n$ : $8$ Transitive number $t$ : $33$ Group : $C_2^4:C_6$ CHM label : $E(8):A_{4}=[1/3.A(4)^{2}]2=E(4):6$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,3)(2,8)(4,6)(5,7), (4,6)(5,7), (1,8)(2,3)(4,5)(6,7), (1,2,3)(4,6,5), (1,5)(2,6)(3,7)(4,8) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: None

## Low degree siblings

8T33, 12T58 x 2, 12T59 x 2, 16T183, 24T181 x 2, 24T182 x 2, 24T183 x 2, 24T184 x 2, 24T185, 24T186, 32T389

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 $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1$ $6$ $2$ $(4,5)(6,7)$ $3, 3, 1, 1$ $16$ $3$ $(2,3,8)(5,7,6)$ $3, 3, 1, 1$ $16$ $3$ $(2,8,3)(5,6,7)$ $2, 2, 2, 2$ $6$ $2$ $(1,2)(3,8)(4,5)(6,7)$ $2, 2, 2, 2$ $3$ $2$ $(1,2)(3,8)(4,7)(5,6)$ $6, 2$ $16$ $6$ $(1,4)(2,5,3,7,8,6)$ $6, 2$ $16$ $6$ $(1,4)(2,6,8,7,3,5)$ $2, 2, 2, 2$ $4$ $2$ $(1,4)(2,7)(3,6)(5,8)$ $4, 4$ $12$ $4$ $(1,4,2,7)(3,6,8,5)$

## Group invariants

 Order: $96=2^{5} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes GAP id: [96, 70]
 Character table:  2 5 4 1 1 4 5 1 1 3 3 3 1 . 1 1 . . 1 1 1 . 1a 2a 3a 3b 2b 2c 6a 6b 2d 4a 2P 1a 1a 3b 3a 1a 1a 3a 3b 1a 2c 3P 1a 2a 1a 1a 2b 2c 2d 2d 2d 4a 5P 1a 2a 3b 3a 2b 2c 6b 6a 2d 4a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 1 -1 -1 -1 -1 X.3 1 1 A /A 1 1 -/A -A -1 -1 X.4 1 1 /A A 1 1 -A -/A -1 -1 X.5 1 1 A /A 1 1 /A A 1 1 X.6 1 1 /A A 1 1 A /A 1 1 X.7 3 -1 . . -1 3 . . -3 1 X.8 3 -1 . . -1 3 . . 3 -1 X.9 6 -2 . . 2 -2 . . . . X.10 6 2 . . -2 -2 . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3