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Properties

Label 12T227
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 $t$ :  $227$
CHM label :  $[2^{6}]S_{4}(6d)=2wrS_{4}(6d)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12), (2,8)(3,9)(4,10)(5,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (2,10)(3,11)(4,8)(5,9)
$|\Aut(F/K)|$:  $2$

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:  $C_2^3:S_4$ x 2, 12T100
384:  16T747
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

12T226 x 8, 12T227 x 7, 24T3089 x 2, 24T3115 x 2, 24T3152 x 2, 24T3206 x 2, 24T3628 x 2, 24T4307 x 2, 24T4877 x 4, 24T4878 x 4, 24T4879 x 4, 24T4880 x 8, 24T4881 x 4, 24T4882 x 4, 24T4883 x 4, 24T4884 x 4, 24T4885 x 8, 24T4886 x 4, 24T4887 x 8, 24T4888 x 4, 24T4889 x 4, 24T4890 x 8, 24T4891 x 4, 24T4892 x 4, 24T4893 x 8, 32T97069 x 8

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 $ $12$ $2$ $( 1,12)( 4, 5)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 1,12)( 4, 5)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 1,12)( 4, 5)( 8, 9)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 1,12)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 1,12)( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 1, 1 $ $6$ $2$ $( 1,12)( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 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 $ $24$ $2$ $( 1,12)( 2, 8)( 3, 9)( 4,10)( 5,11)$
$ 4, 2, 2, 1, 1, 1, 1 $ $24$ $4$ $( 2, 8)( 3, 9)( 4,10, 5,11)$
$ 4, 2, 2, 2, 1, 1 $ $48$ $4$ $( 1,12)( 2, 8)( 3, 9)( 4,10, 5,11)$
$ 4, 4, 1, 1, 1, 1 $ $12$ $4$ $( 2, 9, 3, 8)( 4,10, 5,11)$
$ 4, 4, 2, 1, 1 $ $24$ $4$ $( 1,12)( 2, 9, 3, 8)( 4,10, 5,11)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1,12)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6, 7)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1,12)( 2, 8)( 3, 9)( 4,10, 5,11)( 6, 7)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1,12)( 2, 9, 3, 8)( 4,10, 5,11)( 6, 7)$
$ 3, 3, 3, 3 $ $128$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 6, 3, 3 $ $128$ $6$ $( 1, 5, 9,12, 4, 8)( 2, 6,10)( 3, 7,11)$
$ 6, 3, 3 $ $128$ $6$ $( 1, 5, 9)( 2, 6,11, 3, 7,10)( 4, 8,12)$
$ 6, 6 $ $128$ $6$ $( 1, 5, 9,12, 4, 8)( 2, 6,11, 3, 7,10)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $24$ $2$ $( 2,10)( 3,11)( 4, 8)( 5, 9)$
$ 2, 2, 2, 2, 2, 1, 1 $ $24$ $2$ $( 1,12)( 2,10)( 3,11)( 4, 8)( 5, 9)$
$ 4, 2, 2, 1, 1, 1, 1 $ $48$ $4$ $( 2,10)( 3,11)( 4, 8, 5, 9)$
$ 4, 2, 2, 2, 1, 1 $ $48$ $4$ $( 1,12)( 2,10)( 3,11)( 4, 8, 5, 9)$
$ 4, 4, 1, 1, 1, 1 $ $24$ $4$ $( 2,11, 3,10)( 4, 8, 5, 9)$
$ 4, 4, 2, 1, 1 $ $24$ $4$ $( 1,12)( 2,11, 3,10)( 4, 8, 5, 9)$
$ 2, 2, 2, 2, 2, 1, 1 $ $24$ $2$ $( 2,10)( 3,11)( 4, 8)( 5, 9)( 6, 7)$
$ 2, 2, 2, 2, 2, 2 $ $24$ $2$ $( 1,12)( 2,10)( 3,11)( 4, 8)( 5, 9)( 6, 7)$
$ 4, 2, 2, 2, 1, 1 $ $48$ $4$ $( 2,10)( 3,11)( 4, 8, 5, 9)( 6, 7)$
$ 4, 2, 2, 2, 2 $ $48$ $4$ $( 1,12)( 2,10)( 3,11)( 4, 8, 5, 9)( 6, 7)$
$ 4, 4, 2, 1, 1 $ $24$ $4$ $( 2,11, 3,10)( 4, 8, 5, 9)( 6, 7)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1,12)( 2,11, 3,10)( 4, 8, 5, 9)( 6, 7)$
$ 4, 4, 2, 2 $ $96$ $4$ $( 1, 7)( 2,10, 8, 4)( 3,11, 9, 5)( 6,12)$
$ 4, 4, 4 $ $96$ $4$ $( 1, 7,12, 6)( 2,10, 8, 4)( 3,11, 9, 5)$
$ 8, 2, 2 $ $96$ $8$ $( 1, 7)( 2,10, 8, 5, 3,11, 9, 4)( 6,12)$
$ 8, 4 $ $96$ $8$ $( 1, 7,12, 6)( 2,10, 8, 5, 3,11, 9, 4)$

Group invariants

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