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Properties

Label 12T81
Order \(144\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_6:D_6$

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $81$
Group :  $D_6:D_6$
CHM label :  $[3^{2}:2]D(4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7)(3,9)(5,11), (1,5)(2,10)(4,8)(7,11), (2,6,10)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$ x 2
8:  $D_{4}$ x 2, $C_2^3$
12:  $D_{6}$ x 6
16:  $D_4\times C_2$
24:  $S_3 \times C_2^2$ x 2
36:  $S_3^2$
48:  12T28 x 2
72:  12T37

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: $S_3^2$

Low degree siblings

12T81 x 3, 24T269 x 2, 24T270 x 2, 24T271 x 2, 36T123 x 2, 36T147 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, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 3,11)( 4,12)( 5, 9)( 6,10)$
$ 6, 1, 1, 1, 1, 1, 1 $ $4$ $6$ $( 2, 4, 6, 8,10,12)$
$ 2, 2, 2, 2, 2, 1, 1 $ $18$ $2$ $( 2, 4)( 3,11)( 5, 9)( 6,12)( 8,10)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 4, 8,12)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 2, 8)( 4,10)( 6,12)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 12 $ $12$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$
$ 12 $ $12$ $12$ $( 1, 2, 3,12, 5,10, 7, 8, 9, 6,11, 4)$
$ 6, 6 $ $12$ $6$ $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$
$ 6, 6 $ $12$ $6$ $( 1, 2, 5,10, 9, 6)( 3,12, 7, 8,11, 4)$
$ 4, 4, 4 $ $6$ $4$ $( 1, 2, 7, 8)( 3, 4, 9,10)( 5, 6,11,12)$
$ 4, 4, 4 $ $6$ $4$ $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 3)( 2, 4)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 6, 6 $ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 6, 3, 3 $ $4$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 6,10)( 4, 8,12)$
$ 6, 2, 2, 2 $ $4$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 8)( 4,10)( 6,12)$
$ 6, 3, 3 $ $4$ $6$ $( 1, 3, 5, 7, 9,11)( 2,10, 6)( 4,12, 8)$
$ 6, 6 $ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2,12,10, 8, 6, 4)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 3, 3, 2, 2, 2 $ $4$ $6$ $( 1, 5, 9)( 2, 8)( 3, 7,11)( 4,10)( 6,12)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$

Group invariants

Order:  $144=2^{4} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [144, 154]
Character table: Data not available.