Properties

Label 12T121
Order \(216\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times S_3\wr C_2$

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Group action invariants

Degree $n$:  $12$
Transitive number $t$:  $121$
Group:  $C_3\times S_3\wr C_2$
CHM label:  $1/2[3^{3}:2]dD(4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,8)(4,10)(6,12), (1,7)(3,9)(5,11), (1,2)(3,12)(4,11)(5,10)(6,9)(7,8), (2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
8:  $D_{4}$
12:  $C_6\times C_2$
24:  $D_4 \times C_3$
72:  $C_3^2:D_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: None

Low degree siblings

12T121, 18T93 x 2, 24T561 x 2, 27T84, 36T258 x 2, 36T259 x 2, 36T260 x 2, 36T292 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 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 3, 7,11)( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 3,11, 7)( 4, 8,12)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 2, 4)( 6, 8)(10,12)$
$ 6, 3, 1, 1, 1 $ $12$ $6$ $( 2, 4,10,12, 6, 8)( 3, 7,11)$
$ 6, 3, 1, 1, 1 $ $12$ $6$ $( 2, 4, 6, 8,10,12)( 3,11, 7)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 6, 2, 2, 2 $ $12$ $6$ $( 1, 2)( 3, 4, 7,12,11, 8)( 5,10)( 6, 9)$
$ 6, 2, 2, 2 $ $12$ $6$ $( 1, 2)( 3, 8,11,12, 7, 4)( 5,10)( 6, 9)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 12 $ $18$ $12$ $( 1, 2, 3, 4, 5,10, 7,12, 9, 6,11, 8)$
$ 12 $ $18$ $12$ $( 1, 2, 3, 8, 9, 6,11,12, 5,10, 7, 4)$
$ 4, 4, 4 $ $18$ $4$ $( 1, 2, 3,12)( 4, 9, 6,11)( 5,10, 7, 8)$
$ 6, 6 $ $6$ $6$ $( 1, 2, 5,10, 9, 6)( 3, 8, 7, 4,11,12)$
$ 6, 6 $ $12$ $6$ $( 1, 2, 5,10, 9, 6)( 3,12,11, 4, 7, 8)$
$ 6, 6 $ $6$ $6$ $( 1, 2, 9, 6, 5,10)( 3, 4,11, 8, 7,12)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$
$ 6, 6 $ $9$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4,10,12, 6, 8)$
$ 6, 6 $ $9$ $6$ $( 1, 3, 9,11, 5, 7)( 2, 4, 6, 8,10,12)$
$ 3, 3, 2, 2, 2 $ $12$ $6$ $( 1, 3)( 2, 6,10)( 4,12, 8)( 5, 7)( 9,11)$
$ 6, 3, 3 $ $6$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 6,10)( 4, 8,12)$
$ 6, 3, 3 $ $6$ $6$ $( 1, 3, 9,11, 5, 7)( 2,10, 6)( 4,12, 8)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 9, 5)( 2, 6,10)( 3,11, 7)( 4, 8,12)$

Group invariants

Order:  $216=2^{3} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [216, 157]
Character table: Data not available.