# Properties

 Label 12T126 Order $$288$$ n $$12$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $A_4\wr C_2$

# Related objects

## Group action invariants

 Degree $n$ : $12$ Transitive number $t$ : $126$ Group : $A_4\wr C_2$ CHM label : $[A_{4}^{2}]2=A_{4}wr2$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (2,8)(4,10), (2,6,10)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11) $|\Aut(F/K)|$: $2$

## 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 3: None

Degree 4: None

Degree 6: $S_3\times C_3$

## Low degree siblings

8T42, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $6$ $2$ $( 4,10)( 6,12)$ $2, 2, 2, 2, 1, 1, 1, 1$ $9$ $2$ $( 3, 9)( 4,10)( 5,11)( 6,12)$ $3, 3, 1, 1, 1, 1, 1, 1$ $8$ $3$ $( 2, 4, 6)( 8,10,12)$ $3, 3, 2, 2, 1, 1$ $24$ $6$ $( 2, 4, 6)( 3, 9)( 5,11)( 8,10,12)$ $3, 3, 1, 1, 1, 1, 1, 1$ $8$ $3$ $( 2, 6, 4)( 8,12,10)$ $3, 3, 2, 2, 1, 1$ $24$ $6$ $( 2, 6, 4)( 3, 9)( 5,11)( 8,12,10)$ $2, 2, 2, 2, 2, 2$ $12$ $2$ $( 1, 2)( 3, 4)( 5,12)( 6,11)( 7, 8)( 9,10)$ $4, 4, 2, 2$ $36$ $4$ $( 1, 2)( 3, 4, 9,10)( 5,12,11, 6)( 7, 8)$ $6, 6$ $48$ $6$ $( 1, 2, 3, 4, 5,12)( 6, 7, 8, 9,10,11)$ $6, 6$ $48$ $6$ $( 1, 2, 5,12, 3, 4)( 6, 9,10, 7, 8,11)$ $3, 3, 3, 3$ $16$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12)$ $3, 3, 3, 3$ $32$ $3$ $( 1, 3, 5)( 2, 6, 4)( 7, 9,11)( 8,12,10)$ $3, 3, 3, 3$ $16$ $3$ $( 1, 5, 3)( 2, 6, 4)( 7,11, 9)( 8,12,10)$

## Group invariants

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