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Properties

Label 12T238
Order \(2304\)
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 $t$ :  $238$
CHM label :  $[2^{5}]F_{36}:2_{2}{3^{2}:4}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,3), (6,10)(7,11)(8,9), (2,6,10)(3,7,11), (1,3)(2,12)(4,6,8,10)(5,7,9,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $D_{4}$ x 2, $C_4\times C_2$
16:  $C_2^2:C_4$
72:  $C_3^2:D_4$
144:  12T79
1152:  $S_4\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $C_3^2:D_4$

Low degree siblings

12T237 x 2, 12T238, 16T1496 x 2, 16T1497 x 2, 24T5093 x 2, 24T5117 x 2, 24T5118 x 2, 24T5119, 24T5120 x 2, 24T5121 x 2, 24T5122 x 2, 24T5123 x 2, 24T5124, 24T5125 x 2, 24T5126 x 2, 24T5127 x 2, 24T5128 x 2, 32T205436 x 2, 32T205437 x 2, 32T205438, 32T205439, 36T3213 x 2, 36T3215 x 2, 36T3216 x 2, 36T3218, 36T3224, 36T3449 x 2, 36T3450 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, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1,12)( 2, 3)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 2, 3)( 6, 7)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 1,12)( 2, 3)( 6, 7)( 8, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 2, 3)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $16$ $3$ $( 2,10, 6)( 3,11, 7)$
$ 6, 2, 1, 1, 1, 1 $ $48$ $6$ $( 1,12)( 2,10, 6, 3,11, 7)$
$ 3, 3, 2, 2, 1, 1 $ $48$ $6$ $( 1,12)( 2,10, 7)( 3,11, 6)( 8, 9)$
$ 6, 2, 2, 2 $ $16$ $6$ $( 1,12)( 2,11, 6, 3,10, 7)( 4, 5)( 8, 9)$
$ 3, 3, 3, 3 $ $64$ $3$ $( 1, 5, 9)( 2,10, 6)( 3,11, 7)( 4, 8,12)$
$ 6, 6 $ $64$ $6$ $( 1, 5, 9,12, 4, 8)( 2,10, 6, 3,11, 7)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $36$ $2$ $( 6,10)( 7,11)( 8, 9)$
$ 2, 2, 2, 2, 2, 1, 1 $ $36$ $2$ $( 1,12)( 2, 3)( 6,10)( 7,11)( 8, 9)$
$ 4, 2, 2, 1, 1, 1, 1 $ $36$ $4$ $( 2, 3)( 6,10, 7,11)( 8, 9)$
$ 4, 2, 2, 1, 1, 1, 1 $ $36$ $4$ $( 1,12)( 6,10, 7,11)( 8, 9)$
$ 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $4$ $( 6,10, 7,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 2, 3)( 6,10)( 7,11)$
$ 4, 2, 2, 2, 2 $ $12$ $4$ $( 1,12)( 2, 3)( 4, 5)( 6,11, 7,10)( 8, 9)$
$ 2, 2, 2, 2, 2, 1, 1 $ $12$ $2$ $( 1,12)( 4, 5)( 6,11)( 7,10)( 8, 9)$
$ 6, 2, 2, 1, 1 $ $96$ $6$ $( 1, 5, 9,12, 4, 8)( 6,10)( 7,11)$
$ 3, 3, 2, 2, 2 $ $96$ $6$ $( 1, 5, 9)( 2, 3)( 4, 8,12)( 6,10)( 7,11)$
$ 6, 4, 2 $ $96$ $12$ $( 1, 5, 9,12, 4, 8)( 2, 3)( 6,10, 7,11)$
$ 4, 3, 3, 1, 1 $ $96$ $12$ $( 1, 5, 9)( 4, 8,12)( 6,10, 7,11)$
$ 4, 4, 1, 1, 1, 1 $ $36$ $4$ $( 4, 9, 5, 8)( 6,10, 7,11)$
$ 4, 4, 2, 2 $ $36$ $4$ $( 1,12)( 2, 3)( 4, 9, 5, 8)( 6,10, 7,11)$
$ 4, 2, 2, 2, 1, 1 $ $72$ $4$ $( 2, 3)( 4, 9, 5, 8)( 6,10)( 7,11)$
$ 4, 2, 2, 2, 1, 1 $ $72$ $4$ $( 1,12)( 4, 9, 5, 8)( 6,10)( 7,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $36$ $2$ $( 4, 8)( 5, 9)( 6,10)( 7,11)$
$ 2, 2, 2, 2, 2, 2 $ $36$ $2$ $( 1,12)( 2, 3)( 4, 8)( 5, 9)( 6,10)( 7,11)$
$ 4, 4, 2, 2 $ $144$ $4$ $( 1, 3)( 2,12)( 4, 6, 8,10)( 5, 7, 9,11)$
$ 4, 4, 2, 2 $ $144$ $4$ $( 1, 2)( 3,12)( 4, 6, 8,10)( 5, 7, 9,11)$
$ 8, 4 $ $144$ $8$ $( 1, 2,12, 3)( 4, 7, 9,11, 5, 6, 8,10)$
$ 8, 4 $ $144$ $8$ $( 1, 3,12, 2)( 4, 7, 9,11, 5, 6, 8,10)$
$ 4, 2, 2, 2, 2 $ $72$ $4$ $( 1, 3)( 2,12)( 4,10)( 5,11)( 6, 9, 7, 8)$
$ 4, 2, 2, 2, 2 $ $72$ $4$ $( 1, 2)( 3,12)( 4,10)( 5,11)( 6, 9, 7, 8)$
$ 4, 4, 4 $ $24$ $4$ $( 1, 2,12, 3)( 4,11, 5,10)( 6, 8, 7, 9)$
$ 4, 4, 4 $ $24$ $4$ $( 1, 3,12, 2)( 4,11, 5,10)( 6, 8, 7, 9)$
$ 12 $ $192$ $12$ $( 1,11, 5, 7, 8, 2,12,10, 4, 6, 9, 3)$
$ 12 $ $192$ $12$ $( 1,11, 5, 7, 8, 3,12,10, 4, 6, 9, 2)$

Group invariants

Order:  $2304=2^{8} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.