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Properties

Label 12T131
Order \(324\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times C_3^2:(C_3:C_4)$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $131$
Group :  $C_3\times C_3^2:(C_3:C_4)$
CHM label :  $[3^{4}]4=3wr4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $S_3$, $C_6$
12:  $C_{12}$, $C_3 : C_4$
18:  $S_3\times C_3$
36:  $C_3^2:C_4$, $C_3\times (C_3 : C_4)$
108:  12T72, 12T73

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Low degree siblings

12T131 x 3, 18T123 x 4, 36T497 x 4, 36T514 x 2, 36T527 x 2, 36T532 x 4, 36T536 x 4, 36T543 x 4, 36T544 x 8

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, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 4, 8,12)$
$ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 3, 7,11)( 4, 8,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 3, 7,11)( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 3,11, 7)( 4, 8,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 3,11, 7)( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $2$ $3$ $( 2, 6,10)( 4, 8,12)$
$ 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 $ $4$ $3$ $( 2, 6,10)( 3, 7,11)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 3,11, 7)( 4, 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $2$ $3$ $( 2,10, 6)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2,10, 6)( 3, 7,11)( 4, 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2,10, 6)( 3, 7,11)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2,10, 6)( 3,11, 7)( 4, 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 4, 4, 4 $ $27$ $4$ $( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10,11,12)$
$ 12 $ $27$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$
$ 12 $ $27$ $12$ $( 1, 2, 3, 4, 9,10,11,12, 5, 6, 7, 8)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$
$ 6, 2, 2, 2 $ $18$ $6$ $( 1, 3)( 2, 4, 6, 8,10,12)( 5, 7)( 9,11)$
$ 6, 2, 2, 2 $ $18$ $6$ $( 1, 3)( 2, 4,10,12, 6, 8)( 5, 7)( 9,11)$
$ 6, 6 $ $9$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 6, 6 $ $18$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4,10,12, 6, 8)$
$ 6, 6 $ $9$ $6$ $( 1, 3, 9,11, 5, 7)( 2, 4,10,12, 6, 8)$
$ 4, 4, 4 $ $27$ $4$ $( 1, 4, 3, 2)( 5, 8, 7, 6)( 9,12,11,10)$
$ 12 $ $27$ $12$ $( 1, 4, 7, 6, 5, 8,11,10, 9,12, 3, 2)$
$ 12 $ $27$ $12$ $( 1, 4,11,10, 9,12, 7, 6, 5, 8, 3, 2)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4,12, 8)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 5, 9)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$

Group invariants

Order:  $324=2^{2} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [324, 162]
Character table: Data not available.