# Properties

 Label 12T128 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 $t$ : $128$ Group : $A_4\wr C_2$ CHM label : $[1/4E(4)^{3}:3]S(3)$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,10)(3,12)(4,7)(6,9), (1,7,10)(2,5,11)(3,6,9), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12) $|\Aut(F/K)|$: $1$

## 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: None

Degree 3: $S_3$

Degree 4: None

Degree 6: None

## Low degree siblings

8T42, 12T126, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 1, 1, 1$ $16$ $3$ $( 4, 7,10)( 5,11, 8)( 6, 9,12)$ $3, 3, 3, 1, 1, 1$ $16$ $3$ $( 4,10, 7)( 5, 8,11)( 6,12, 9)$ $2, 2, 2, 2, 1, 1, 1, 1$ $12$ $2$ $( 2, 3)( 5,12)( 6,11)( 8, 9)$ $6, 3, 2, 1$ $48$ $6$ $( 2, 3)( 4, 7,10)( 5, 6, 8,12,11, 9)$ $6, 3, 2, 1$ $48$ $6$ $( 2, 3)( 4,10, 7)( 5, 9,11,12, 8, 6)$ $2, 2, 2, 2, 1, 1, 1, 1$ $9$ $2$ $( 2, 5)( 3,12)( 6, 9)( 8,11)$ $3, 3, 3, 3$ $8$ $3$ $( 1, 2, 3)( 4, 5, 9)( 6,10, 8)( 7,11,12)$ $3, 3, 3, 3$ $8$ $3$ $( 1, 2, 3)( 4, 8,12)( 5, 6, 7)( 9,10,11)$ $3, 3, 3, 3$ $32$ $3$ $( 1, 2, 3)( 4,11, 6)( 5,12,10)( 7, 8, 9)$ $4, 4, 2, 2$ $36$ $4$ $( 1, 2, 4,11)( 3, 6)( 5, 7, 8,10)( 9,12)$ $6, 6$ $24$ $6$ $( 1, 2, 6, 7,11, 9)( 3, 4, 5,12,10, 8)$ $6, 6$ $24$ $6$ $( 1, 2, 6,10,11,12)( 3, 4, 8, 9, 7, 5)$ $2, 2, 2, 2, 2, 2$ $6$ $2$ $( 1, 4)( 2, 5)( 3, 9)( 6,12)( 7,10)( 8,11)$

## Group invariants

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