# Properties

 Label 8T42 Order $$288$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $A_4\wr C_2$

# Related objects

## Group action invariants

 Degree $n$ : $8$ Transitive number $t$ : $42$ Group : $A_4\wr C_2$ CHM label : $[A(4)^{2}]2$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,3)(2,8), (1,2,3), (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:  $S_3$, $C_6$
18:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: None

## Low degree siblings

12T126, 12T128, 12T129, 16T708, 18T112, 18T113, 24T692, 24T694, 24T695, 24T702, 24T703, 24T704, 32T9306, 36T316, 36T318, 36T456, 36T457, 36T458, 36T459

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 $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 1, 1, 1, 1, 1$ $8$ $3$ $(5,6,7)$ $3, 1, 1, 1, 1, 1$ $8$ $3$ $(5,7,6)$ $2, 2, 1, 1, 1, 1$ $6$ $2$ $(4,5)(6,7)$ $3, 3, 1, 1$ $32$ $3$ $(2,3,8)(5,6,7)$ $3, 3, 1, 1$ $16$ $3$ $(2,3,8)(5,7,6)$ $3, 2, 2, 1$ $24$ $6$ $(2,3,8)(4,5)(6,7)$ $3, 3, 1, 1$ $16$ $3$ $(2,8,3)(5,6,7)$ $3, 2, 2, 1$ $24$ $6$ $(2,8,3)(4,5)(6,7)$ $2, 2, 2, 2$ $9$ $2$ $(1,2)(3,8)(4,5)(6,7)$ $2, 2, 2, 2$ $12$ $2$ $(1,4)(2,5)(3,7)(6,8)$ $6, 2$ $48$ $6$ $(1,4)(2,5,3,7,8,6)$ $6, 2$ $48$ $6$ $(1,4)(2,5,8,6,3,7)$ $4, 4$ $36$ $4$ $(1,4,2,5)(3,7,8,6)$

## Group invariants

 Order: $288=2^{5} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [288, 1025]
 Character table:  2 5 2 2 4 . 1 2 1 2 5 3 1 1 3 3 2 2 2 1 2 2 1 2 1 . 1 1 1 . 1a 3a 3b 2a 3c 3d 6a 3e 6b 2b 2c 6c 6d 4a 2P 1a 3b 3a 1a 3c 3e 3a 3d 3b 1a 1a 3d 3e 2b 3P 1a 1a 1a 2a 1a 1a 2a 1a 2a 2b 2c 2c 2c 4a 5P 1a 3b 3a 2a 3c 3e 6b 3d 6a 2b 2c 6d 6c 4a X.1 1 1 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 -1 -1 -1 X.3 1 A /A 1 1 A /A /A A 1 -1 -/A -A -1 X.4 1 /A A 1 1 /A A A /A 1 -1 -A -/A -1 X.5 1 A /A 1 1 A /A /A A 1 1 /A A 1 X.6 1 /A A 1 1 /A A A /A 1 1 A /A 1 X.7 2 -1 -1 2 -1 2 -1 2 -1 2 . . . . X.8 2 -A -/A 2 -1 C -/A /C -A 2 . . . . X.9 2 -/A -A 2 -1 /C -A C -/A 2 . . . . X.10 6 3 3 2 . . -1 . -1 -2 . . . . X.11 6 B /B 2 . . -/A . -A -2 . . . . X.12 6 /B B 2 . . -A . -/A -2 . . . . X.13 9 . . -3 . . . . . 1 -3 . . 1 X.14 9 . . -3 . . . . . 1 3 . . -1 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 C = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3