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Properties

Label 12T261
Order \(5184\)
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 $t$ :  $261$
CHM label :  $[S(3)^{4}]E(4)=S(3)wrE(4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (4,8), (4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)
$|\Aut(F/K)|$:  $1$

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 6, $C_2^3$
16:  $D_4\times C_2$ x 3
32:  $C_2^2 \wr C_2$
64:  $(((C_4 \times C_2): C_2):C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: None

Low degree siblings

12T267 x 3, 18T481 x 3, 24T7723, 24T7724 x 3, 24T7725 x 3, 24T7726 x 3, 24T7727 x 3, 24T7728, 24T7729, 24T7762 x 3, 24T7763 x 3, 24T7764 x 3, 24T7765 x 3, 24T7766 x 3, 24T7767 x 3, 24T7768 x 3, 24T7782, 36T6224 x 3, 36T6225 x 3, 36T6226 x 3, 36T6227 x 3, 36T6228 x 3, 36T6229 x 3, 36T6230 x 3, 36T6290 x 3

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, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 3,11, 7)( 4, 8,12)$
$ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 4, 8,12)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 9, 5)( 2, 6,10)( 3,11, 7)( 4, 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $32$ $3$ $( 1, 9, 5)( 2, 6,10)( 4, 8,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 2, 6,10)( 4, 8,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 2, 6,10)( 3, 7,11)$
$ 2, 2, 2, 2, 2, 2 $ $36$ $2$ $( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)$
$ 6, 2, 2, 2 $ $144$ $6$ $( 1,10)( 2, 5)( 3, 4, 7, 8,11,12)( 6, 9)$
$ 6, 6 $ $144$ $6$ $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 7,11)( 8,12)$
$ 3, 3, 2, 2, 1, 1 $ $72$ $6$ $( 1, 9, 5)( 2, 6,10)( 7,11)( 8,12)$
$ 3, 2, 2, 1, 1, 1, 1, 1 $ $72$ $6$ $( 2, 6,10)( 7,11)( 8,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $81$ $2$ $( 5, 9)( 6,10)( 7,11)( 8,12)$
$ 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 8,12)$
$ 3, 2, 1, 1, 1, 1, 1, 1, 1 $ $24$ $6$ $( 3,11, 7)( 4, 8)$
$ 3, 3, 2, 1, 1, 1, 1 $ $48$ $6$ $( 1, 9, 5)( 2, 6,10)( 8,12)$
$ 3, 3, 3, 2, 1 $ $96$ $6$ $( 1, 9, 5)( 2, 6,10)( 3,11, 7)( 4, 8)$
$ 3, 2, 1, 1, 1, 1, 1, 1, 1 $ $24$ $6$ $( 2, 6,10)( 8,12)$
$ 3, 3, 2, 1, 1, 1, 1 $ $48$ $6$ $( 2, 6,10)( 3,11, 7)( 4, 8)$
$ 3, 2, 1, 1, 1, 1, 1, 1, 1 $ $24$ $6$ $( 1, 5, 9)( 8,12)$
$ 3, 3, 2, 1, 1, 1, 1 $ $48$ $6$ $( 1, 5, 9)( 3,11, 7)( 4, 8)$
$ 4, 2, 2, 2, 2 $ $216$ $4$ $( 1,10)( 2, 5)( 3,12,11, 8)( 4, 7)( 6, 9)$
$ 6, 4, 2 $ $432$ $12$ $( 1, 2, 5, 6, 9,10)( 3,12,11, 8)( 4, 7)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $108$ $2$ $( 5, 9)( 6,10)( 8,12)$
$ 3, 2, 2, 2, 1, 1, 1 $ $216$ $6$ $( 3,11, 7)( 4, 8)( 5, 9)( 6,10)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 6,10)( 8,12)$
$ 3, 2, 2, 1, 1, 1, 1, 1 $ $72$ $6$ $( 3,11, 7)( 4, 8)( 6,10)$
$ 3, 3, 2, 2, 1, 1 $ $72$ $6$ $( 1, 9, 5)( 2, 6)( 3,11, 7)( 4, 8)$
$ 4, 4, 2, 2 $ $324$ $4$ $( 1,10, 9, 6)( 2, 5)( 3,12,11, 8)( 4, 7)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 6,10)( 7,11)$
$ 3, 2, 2, 1, 1, 1, 1, 1 $ $72$ $6$ $( 3,11)( 4, 8,12)( 6,10)$
$ 3, 3, 2, 2, 1, 1 $ $72$ $6$ $( 1, 9, 5)( 2, 6)( 3,11)( 4, 8,12)$
$ 2, 2, 2, 2, 2, 2 $ $36$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 6, 6 $ $144$ $6$ $( 1, 3, 9,11, 5, 7)( 2,12, 6, 4,10, 8)$
$ 6, 2, 2, 2 $ $144$ $6$ $( 1, 7)( 2,12, 6, 4,10, 8)( 3, 9)( 5,11)$
$ 2, 2, 2, 2, 2, 2 $ $36$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
$ 6, 6 $ $144$ $6$ $( 1, 8, 5,12, 9, 4)( 2, 7,10, 3, 6,11)$
$ 6, 2, 2, 2 $ $144$ $6$ $( 1, 8, 5,12, 9, 4)( 2,11)( 3, 6)( 7,10)$
$ 4, 4, 2, 2 $ $324$ $4$ $( 1,11, 5, 7)( 2,12, 6, 8)( 3, 9)( 4,10)$
$ 4, 4, 2, 2 $ $324$ $4$ $( 1, 8, 5, 4)( 2, 3, 6,11)( 7,10)( 9,12)$
$ 4, 2, 2, 2, 2 $ $216$ $4$ $( 1, 7)( 2,12, 6, 8)( 3, 9)( 4,10)( 5,11)$
$ 6, 4, 2 $ $432$ $12$ $( 1, 3, 9,11, 5, 7)( 2, 4,10, 8)( 6,12)$
$ 4, 2, 2, 2, 2 $ $216$ $4$ $( 1, 4)( 2, 3, 6,11)( 5, 8)( 7,10)( 9,12)$
$ 6, 4, 2 $ $432$ $12$ $( 1, 8, 5,12, 9, 4)( 2,11)( 3, 6, 7,10)$

Group invariants

Order:  $5184=2^{6} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.