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Properties

Label 12T224
Order \(1536\)
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 $t$ :  $224$
CHM label :  $[2^{6}]S_{4}(6c)=2wrS_{4}(6c)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12), (2,8)(3,9)(4,10)(5,11), (1,7)(2,10)(3,11)(4,8)(5,9)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
24:  $S_4$ x 3
48:  $S_4\times C_2$ x 3
96:  $V_4^2:S_3$
192:  12T100
384:  16T751
768:  16T1063

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$

Low degree siblings

12T223 x 4, 12T224 x 3, 24T2996 x 2, 24T3106, 24T3107, 24T3112 x 2, 24T3117, 24T3122, 24T3149, 24T3170, 24T3176, 24T3180, 24T3560, 24T3600, 24T4285 x 2, 24T4287, 24T4357, 24T4857 x 2, 24T4858 x 2, 24T4859 x 4, 24T4860 x 2, 24T4861 x 2, 24T4862 x 2, 24T4863 x 2, 24T4864 x 2, 24T4865 x 2, 24T4866 x 2, 24T4867 x 2, 24T4868 x 2, 24T4869 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 $ $6$ $2$ $( 1,12)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 1,12)( 6, 7)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 1,12)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 1,12)( 6, 7)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 1,12)( 4, 5)( 6, 7)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 1,12)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 1,12)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 1, 1 $ $6$ $2$ $( 1,12)( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 2, 8)( 3, 9)( 4,10)( 5,11)$
$ 2, 2, 2, 2, 2, 1, 1 $ $24$ $2$ $( 1,12)( 2, 8)( 3, 9)( 4,10)( 5,11)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1,12)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6, 7)$
$ 4, 2, 2, 1, 1, 1, 1 $ $24$ $4$ $( 2, 8)( 3, 9)( 4,11, 5,10)$
$ 4, 2, 2, 2, 1, 1 $ $48$ $4$ $( 1,12)( 2, 8)( 3, 9)( 4,11, 5,10)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1,12)( 2, 8)( 3, 9)( 4,11, 5,10)( 6, 7)$
$ 4, 4, 1, 1, 1, 1 $ $12$ $4$ $( 2, 9, 3, 8)( 4,11, 5,10)$
$ 4, 4, 2, 1, 1 $ $24$ $4$ $( 1,12)( 2, 9, 3, 8)( 4,11, 5,10)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1,12)( 2, 9, 3, 8)( 4,11, 5,10)( 6, 7)$
$ 3, 3, 3, 3 $ $128$ $3$ $( 1, 3, 5)( 2, 4,12)( 6, 8,10)( 7, 9,11)$
$ 6, 3, 3 $ $128$ $6$ $( 1, 3, 5,12, 2, 4)( 6, 8,10)( 7, 9,11)$
$ 6, 3, 3 $ $128$ $6$ $( 1, 3, 5)( 2, 4,12)( 6, 8,10, 7, 9,11)$
$ 6, 6 $ $128$ $6$ $( 1, 3, 5,12, 2, 4)( 6, 8,10, 7, 9,11)$
$ 4, 4, 1, 1, 1, 1 $ $48$ $4$ $( 2,10, 8, 4)( 3,11, 9, 5)$
$ 4, 4, 2, 1, 1 $ $48$ $4$ $( 1,12)( 2,10, 8, 4)( 3,11, 9, 5)$
$ 4, 4, 2, 1, 1 $ $48$ $4$ $( 2,10, 8, 4)( 3,11, 9, 5)( 6, 7)$
$ 4, 4, 2, 2 $ $48$ $4$ $( 1,12)( 2,10, 8, 4)( 3,11, 9, 5)( 6, 7)$
$ 8, 1, 1, 1, 1 $ $48$ $8$ $( 2,11, 9, 5, 3,10, 8, 4)$
$ 8, 2, 1, 1 $ $48$ $8$ $( 1,12)( 2,11, 9, 5, 3,10, 8, 4)$
$ 8, 2, 1, 1 $ $48$ $8$ $( 2,11, 9, 5, 3,10, 8, 4)( 6, 7)$
$ 8, 2, 2 $ $48$ $8$ $( 1,12)( 2,11, 9, 5, 3,10, 8, 4)( 6, 7)$
$ 2, 2, 2, 2, 2, 2 $ $48$ $2$ $( 1, 7)( 2,10)( 3,11)( 4, 8)( 5, 9)( 6,12)$
$ 4, 2, 2, 2, 2 $ $48$ $4$ $( 1, 7,12, 6)( 2,10)( 3,11)( 4, 8)( 5, 9)$
$ 4, 2, 2, 2, 2 $ $96$ $4$ $( 1, 7)( 2,11, 3,10)( 4, 8)( 5, 9)( 6,12)$
$ 4, 4, 2, 2 $ $96$ $4$ $( 1, 7,12, 6)( 2,11, 3,10)( 4, 8)( 5, 9)$
$ 4, 4, 2, 2 $ $48$ $4$ $( 1, 7)( 2,11, 3,10)( 4, 8, 5, 9)( 6,12)$
$ 4, 4, 4 $ $48$ $4$ $( 1, 7,12, 6)( 2,11, 3,10)( 4, 8, 5, 9)$

Group invariants

Order:  $1536=2^{9} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.