# Properties

 Label 12T208 Order $$1152$$ n $$12$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No

# Related objects

## Group action invariants

 Degree $n$ : $12$ Transitive number $t$ : $208$ CHM label : $[2A_{4}^{2}]2=2A4wr2=2wrF_{18}(6)$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (6,12), (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:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$, $C_6$ x 3
8:  $D_{4}$
12:  $D_{6}$, $C_6\times C_2$
18:  $S_3\times C_3$
24:  $(C_6\times C_2):C_2$, $D_4 \times C_3$
36:  $C_6\times S_3$
72:  12T42
288:  $A_4\wr C_2$
576:  12T158

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

12T208, 16T1286 x 2, 24T2815 x 2, 24T2816 x 2, 24T2817 x 2, 24T2818 x 2, 24T2819, 32T96688, 36T1607, 36T1613 x 2, 36T1635, 36T1914 x 2, 36T1915 x 2

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $6$ $2$ $( 6,12)$ $2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $6$ $2$ $( 4,10)( 6,12)$ $2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $9$ $2$ $( 5,11)( 6,12)$ $2, 2, 2, 1, 1, 1, 1, 1, 1$ $18$ $2$ $( 4,10)( 5,11)( 6,12)$ $2, 2, 2, 1, 1, 1, 1, 1, 1$ $2$ $2$ $( 2, 8)( 4,10)( 6,12)$ $2, 2, 2, 2, 1, 1, 1, 1$ $6$ $2$ $( 2, 8)( 4,10)( 5,11)( 6,12)$ $2, 2, 2, 2, 1, 1, 1, 1$ $9$ $2$ $( 3, 9)( 4,10)( 5,11)( 6,12)$ $2, 2, 2, 2, 2, 1, 1$ $6$ $2$ $( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ $2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ $3, 3, 1, 1, 1, 1, 1, 1$ $8$ $3$ $( 2, 6,10)( 4, 8,12)$ $6, 1, 1, 1, 1, 1, 1$ $8$ $6$ $( 2,12, 4, 8, 6,10)$ $3, 3, 2, 1, 1, 1, 1$ $24$ $6$ $( 2, 6,10)( 4, 8,12)( 5,11)$ $6, 2, 1, 1, 1, 1$ $24$ $6$ $( 2,12, 4, 8, 6,10)( 5,11)$ $3, 3, 2, 2, 1, 1$ $24$ $6$ $( 2, 6,10)( 3, 9)( 4, 8,12)( 5,11)$ $6, 2, 2, 1, 1$ $24$ $6$ $( 2,12, 4, 8, 6,10)( 3, 9)( 5,11)$ $3, 3, 2, 2, 2$ $8$ $6$ $( 1, 7)( 2, 6,10)( 3, 9)( 4, 8,12)( 5,11)$ $6, 2, 2, 2$ $8$ $6$ $( 1, 7)( 2,12, 4, 8, 6,10)( 3, 9)( 5,11)$ $3, 3, 1, 1, 1, 1, 1, 1$ $8$ $3$ $( 2,10, 6)( 4,12, 8)$ $6, 1, 1, 1, 1, 1, 1$ $8$ $6$ $( 2,10,12, 8, 4, 6)$ $3, 3, 2, 1, 1, 1, 1$ $24$ $6$ $( 2,10, 6)( 4,12, 8)( 5,11)$ $6, 2, 1, 1, 1, 1$ $24$ $6$ $( 2,10,12, 8, 4, 6)( 5,11)$ $3, 3, 2, 2, 1, 1$ $24$ $6$ $( 2,10, 6)( 3, 9)( 4,12, 8)( 5,11)$ $6, 2, 2, 1, 1$ $24$ $6$ $( 2,10,12, 8, 4, 6)( 3, 9)( 5,11)$ $3, 3, 2, 2, 2$ $8$ $6$ $( 1, 7)( 2,10, 6)( 3, 9)( 4,12, 8)( 5,11)$ $6, 2, 2, 2$ $8$ $6$ $( 1, 7)( 2,10,12, 8, 4, 6)( 3, 9)( 5,11)$ $3, 3, 3, 3$ $16$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ $6, 3, 3$ $32$ $6$ $( 1, 5, 9)( 2,12, 4, 8, 6,10)( 3, 7,11)$ $6, 6$ $16$ $6$ $( 1,11, 3, 7, 5, 9)( 2,12, 4, 8, 6,10)$ $3, 3, 3, 3$ $32$ $3$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$ $6, 3, 3$ $32$ $6$ $( 1, 5, 9)( 2,10,12, 8, 4, 6)( 3, 7,11)$ $6, 3, 3$ $32$ $6$ $( 1,11, 3, 7, 5, 9)( 2,10, 6)( 4,12, 8)$ $6, 6$ $32$ $6$ $( 1,11, 3, 7, 5, 9)( 2,10,12, 8, 4, 6)$ $3, 3, 3, 3$ $16$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ $6, 3, 3$ $32$ $6$ $( 1, 9, 5)( 2,10,12, 8, 4, 6)( 3,11, 7)$ $6, 6$ $16$ $6$ $( 1, 9,11, 7, 3, 5)( 2,10,12, 8, 4, 6)$ $2, 2, 2, 2, 2, 2$ $24$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$ $4, 2, 2, 2, 2$ $72$ $4$ $( 1, 6, 7,12)( 2, 3)( 4, 5)( 8, 9)(10,11)$ $4, 4, 2, 2$ $72$ $4$ $( 1, 6, 7,12)( 2, 3)( 4, 5,10,11)( 8, 9)$ $4, 4, 4$ $24$ $4$ $( 1, 6, 7,12)( 2, 3, 8, 9)( 4, 5,10,11)$ $6, 6$ $96$ $6$ $( 1, 4, 5, 8, 9,12)( 2, 3, 6, 7,10,11)$ $12$ $96$ $12$ $( 1, 4, 5, 8, 9, 6, 7,10,11, 2, 3,12)$ $6, 6$ $96$ $6$ $( 1, 8, 9, 4, 5,12)( 2, 3,10,11, 6, 7)$ $12$ $96$ $12$ $( 1, 8, 9, 4, 5, 6, 7, 2, 3,10,11,12)$

## Group invariants

 Order: $1152=2^{7} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: Data not available
 Character table: Data not available.