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Properties

Label 12T236
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 $t$ :  $236$
CHM label :  $[2^{5}]F_{36}(6):2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,3), (1,5,9)(4,8,12), (4,8)(5,9), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$
72:  $C_3^2:D_4$
144:  12T77
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

12T235 x 2, 12T236, 16T1493 x 2, 16T1494 x 2, 18T370 x 4, 24T5094 x 2, 24T5105, 24T5106 x 2, 24T5107 x 2, 24T5108 x 2, 24T5109 x 2, 24T5110 x 2, 24T5111 x 2, 24T5112, 24T5113 x 2, 24T5114 x 2, 24T5115 x 2, 24T5116 x 2, 32T205429 x 2, 32T205430 x 2, 32T205431, 32T205432, 32T205477 x 4, 36T3212 x 2, 36T3214 x 2, 36T3217, 36T3219 x 2, 36T3221, 36T3223 x 2, 36T3227 x 2, 36T3283 x 4, 36T3284 x 4, 36T3285 x 4, 36T3286 x 4, 36T3448 x 2, 36T3451 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, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 1,12)( 2, 3)( 6, 7)( 8, 9)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 4, 5)( 8, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 1,12)( 2, 3)( 4, 5)( 8, 9)$
$ 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 $ $16$ $3$ $( 1, 5, 9)( 4, 8,12)$
$ 6, 2, 1, 1, 1, 1 $ $48$ $6$ $( 1, 5, 9,12, 4, 8)( 2, 3)$
$ 3, 3, 2, 2, 1, 1 $ $48$ $6$ $( 1, 5, 8)( 2, 3)( 4, 9,12)( 6, 7)$
$ 6, 2, 2, 2 $ $16$ $6$ $( 1, 5, 9,12, 4, 8)( 2, 3)( 6, 7)(10,11)$
$ 3, 3, 3, 3 $ $64$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 6, 6 $ $64$ $6$ $( 1, 5, 9,12, 4, 8)( 2, 6,10, 3, 7,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $(4,8)(5,9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $36$ $2$ $( 1,12)( 2, 3)( 4, 8)( 5, 9)$
$ 4, 2, 1, 1, 1, 1, 1, 1 $ $36$ $4$ $( 4, 9, 5, 8)( 6, 7)$
$ 4, 2, 2, 2, 1, 1 $ $36$ $4$ $( 1,12)( 2, 3)( 4, 9, 5, 8)( 6, 7)$
$ 4, 2, 1, 1, 1, 1, 1, 1 $ $12$ $4$ $( 1,12)( 4, 9, 5, 8)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $36$ $2$ $( 2, 3)( 4, 8)( 5, 9)( 6, 7)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1,12)( 2, 3)( 4, 8)( 5, 9)( 6, 7)(10,11)$
$ 4, 2, 2, 2, 1, 1 $ $12$ $4$ $( 2, 3)( 4, 9, 5, 8)( 6, 7)(10,11)$
$ 3, 3, 2, 2, 1, 1 $ $96$ $6$ $( 2, 6,10)( 3, 7,11)( 4, 8)( 5, 9)$
$ 6, 2, 2, 2 $ $96$ $6$ $( 1,12)( 2, 6,10, 3, 7,11)( 4, 8)( 5, 9)$
$ 6, 4, 1, 1 $ $96$ $12$ $( 2, 7,11, 3, 6,10)( 4, 9, 5, 8)$
$ 4, 3, 3, 2 $ $96$ $12$ $( 1,12)( 2, 7,11)( 3, 6,10)( 4, 9, 5, 8)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $36$ $2$ $( 2,10)( 3,11)( 4, 8)( 5, 9)$
$ 4, 2, 2, 2, 1, 1 $ $72$ $4$ $( 1,12)( 2,10, 3,11)( 4, 8)( 5, 9)$
$ 4, 4, 2, 2 $ $36$ $4$ $( 1,12)( 2,10, 3,11)( 4, 9, 5, 8)( 6, 7)$
$ 4, 4, 1, 1, 1, 1 $ $36$ $4$ $( 2,10, 3,11)( 4, 9, 5, 8)$
$ 4, 2, 2, 2, 1, 1 $ $72$ $4$ $( 1,12)( 2,10)( 3,11)( 4, 9, 5, 8)$
$ 2, 2, 2, 2, 2, 2 $ $36$ $2$ $( 1,12)( 2,10)( 3,11)( 4, 8)( 5, 9)( 6, 7)$
$ 2, 2, 2, 2, 2, 2 $ $24$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 4, 4, 2, 2 $ $72$ $4$ $( 1, 7,12, 6)( 2, 8, 3, 9)( 4,10)( 5,11)$
$ 4, 4, 2, 2 $ $72$ $4$ $( 1, 7)( 2, 9, 3, 8)( 4,10, 5,11)( 6,12)$
$ 2, 2, 2, 2, 2, 2 $ $24$ $2$ $( 1, 7)( 2, 9)( 3, 8)( 4,10)( 5,11)( 6,12)$
$ 6, 6 $ $192$ $6$ $( 1, 7, 5,11, 9, 3)( 2,12, 6, 4,10, 8)$
$ 6, 6 $ $192$ $6$ $( 1, 7, 5,11, 9, 2)( 3,12, 6, 4,10, 8)$
$ 4, 4, 2, 2 $ $144$ $4$ $( 1, 7)( 2, 4,10, 8)( 3, 5,11, 9)( 6,12)$
$ 8, 4 $ $144$ $8$ $( 1, 7,12, 6)( 2, 4,10, 8, 3, 5,11, 9)$
$ 4, 4, 2, 2 $ $144$ $4$ $( 1, 7)( 2, 4,10, 9)( 3, 5,11, 8)( 6,12)$
$ 8, 4 $ $144$ $8$ $( 1, 7,12, 6)( 2, 4,10, 9, 3, 5,11, 8)$

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.