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Properties

Label 12T281
Order \(15552\)
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 $t$ :  $281$
CHM label :  $[3^{4}:2^{3}]S(4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (4,8)(7,11), (2,11)(3,6)(7,10), (4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
24:  $S_4$ x 3
96:  $V_4^2:S_3$
192:  $C_2^3:S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: $S_4$

Degree 6: None

Low degree siblings

12T281 x 2, 24T12156 x 3, 24T12157 x 3, 24T12158 x 3, 24T12163 x 3, 36T10170 x 3, 36T10209

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 $ $1$ $1$ $()$
$ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $24$ $3$ $( 3,11, 7)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $32$ $3$ $( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4, 8,12)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $54$ $2$ $( 7,11)( 8,12)$
$ 3, 2, 2, 1, 1, 1, 1, 1 $ $216$ $6$ $( 2,10, 6)( 7,11)( 8,12)$
$ 3, 3, 2, 2, 1, 1 $ $216$ $6$ $( 1, 9, 5)( 2,10, 6)( 7,11)( 8,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $81$ $2$ $( 5, 9)( 6,10)( 7,11)( 8,12)$
$ 2, 2, 2, 2, 2, 2 $ $54$ $2$ $( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)$
$ 6, 2, 2, 2 $ $216$ $6$ $( 1,10)( 2, 5)( 3, 8,11, 4, 7,12)( 6, 9)$
$ 6, 6 $ $216$ $6$ $( 1, 6, 9, 2, 5,10)( 3, 8,11, 4, 7,12)$
$ 6, 2, 2, 2 $ $216$ $6$ $( 1,10)( 2, 5)( 3, 8, 7, 4,11,12)( 6, 9)$
$ 2, 2, 2, 2, 2, 2 $ $54$ $2$ $( 1,10)( 2, 5)( 3,12)( 4,11)( 6, 9)( 7, 8)$
$ 6, 6 $ $216$ $6$ $( 1, 6, 9, 2, 5,10)( 3, 8, 7, 4,11,12)$
$ 4, 4, 2, 2 $ $972$ $4$ $( 1, 6, 9,10)( 2, 5)( 3, 8,11,12)( 4, 7)$
$ 3, 3, 3, 1, 1, 1 $ $288$ $3$ $( 1,10, 7)( 2,11, 5)( 3, 9, 6)$
$ 3, 3, 3, 3 $ $576$ $3$ $( 1,10, 7)( 2,11, 5)( 3, 9, 6)( 4,12, 8)$
$ 9, 1, 1, 1 $ $576$ $9$ $( 1,10, 3, 9, 6,11, 5, 2, 7)$
$ 9, 3 $ $576$ $9$ $( 1,10, 3, 9, 6,11, 5, 2, 7)( 4,12, 8)$
$ 9, 3 $ $576$ $9$ $( 1,10, 3, 9, 6,11, 5, 2, 7)( 4, 8,12)$
$ 6, 3, 2, 1 $ $2592$ $6$ $( 1,10,11, 5, 2, 7)( 3, 9, 6)( 8,12)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $36$ $2$ $( 1,10)( 2, 5)( 6, 9)$
$ 3, 2, 2, 2, 1, 1, 1 $ $144$ $6$ $( 1,10)( 2, 5)( 4,12, 8)( 6, 9)$
$ 3, 3, 2, 2, 2 $ $72$ $6$ $( 1,10)( 2, 5)( 3,11, 7)( 4,12, 8)( 6, 9)$
$ 3, 3, 2, 2, 2 $ $72$ $6$ $( 1,10)( 2, 5)( 3,11, 7)( 4, 8,12)( 6, 9)$
$ 6, 1, 1, 1, 1, 1, 1 $ $72$ $6$ $( 1, 6, 9, 2, 5,10)$
$ 6, 3, 1, 1, 1 $ $288$ $6$ $( 1, 6, 9, 2, 5,10)( 4,12, 8)$
$ 6, 3, 3 $ $144$ $6$ $( 1, 6, 9, 2, 5,10)( 3,11, 7)( 4,12, 8)$
$ 6, 3, 3 $ $144$ $6$ $( 1, 6, 9, 2, 5,10)( 3,11, 7)( 4, 8,12)$
$ 2, 2, 2, 2, 2, 1, 1 $ $324$ $2$ $( 1,10)( 2, 5)( 6, 9)( 7,11)( 8,12)$
$ 6, 2, 2, 1, 1 $ $648$ $6$ $( 1, 6, 9, 2, 5,10)( 7,11)( 8,12)$
$ 4, 2, 2, 1, 1, 1, 1 $ $648$ $4$ $( 1, 6, 9,10)( 2, 5)( 8,12)$
$ 4, 3, 2, 2, 1 $ $1296$ $12$ $( 1, 6, 9,10)( 2, 5)( 3,11, 7)( 8,12)$
$ 4, 4, 4 $ $648$ $4$ $( 1, 7,10, 4)( 2, 8, 5,11)( 3, 6,12, 9)$
$ 12 $ $1296$ $12$ $( 1, 7,10,12, 9, 3, 6, 8, 5,11, 2, 4)$
$ 12 $ $1296$ $12$ $( 1, 7, 6, 8, 5,11, 2,12, 9, 3,10, 4)$
$ 4, 4, 4 $ $648$ $4$ $( 1, 7, 6, 4)( 2, 8, 5,11)( 3,10,12, 9)$

Group invariants

Order:  $15552=2^{6} \cdot 3^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.