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Properties

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

Related objects

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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 TypeSizeOrderRepresentative
$ 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.