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Properties

Label 12T221
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 $t$ :  $221$
CHM label :  $[1/2.cD(4)^{3}]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,6,9,12), (1,5)(2,10)(4,8)(7,11), (1,7)(3,9), (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:  16T763
768:  24T2216

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4\times C_2$

Low degree siblings

12T221 x 7, 24T3114 x 2, 24T3116 x 4, 24T3120 x 2, 24T3146 x 2, 24T3179 x 2, 24T4298 x 2, 24T4305 x 2, 24T4344 x 2, 24T4831 x 4, 24T4832 x 4, 24T4833 x 4, 24T4834 x 4, 24T4835 x 4, 24T4836 x 4, 24T4837 x 4

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

Group invariants

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