Properties

Label 12T119
Order \(216\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times C_3:S_3.C_2$

Related objects

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Group action invariants

Degree $n$:  $12$
Transitive number $t$:  $119$
Group:  $S_3\times C_3:S_3.C_2$
CHM label:  $[3^{3}:2]4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5)(2,10)(4,8)(7,11), (2,6,10)(3,7,11)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$
8:  $C_4\times C_2$
12:  $D_{6}$
24:  $S_3 \times C_4$
36:  $C_3^2:C_4$
72:  12T40

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Low degree siblings

12T119, 18T95 x 2, 24T559 x 2, 27T85, 36T261 x 2, 36T262 x 2, 36T263 x 2, 36T295 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 $ $27$ $2$ $( 4, 8)( 5, 9)( 6,10)( 7,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 3, 7,11)( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $8$ $3$ $( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 4, 4, 4 $ $27$ $4$ $( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10,11,12)$
$ 12 $ $18$ $12$ $( 1, 2, 3, 4, 5,10, 7,12, 9, 6,11, 8)$
$ 4, 4, 4 $ $9$ $4$ $( 1, 2, 3,12)( 4, 9, 6,11)( 5,10, 7, 8)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$
$ 6, 2, 2, 2 $ $12$ $6$ $( 1, 3)( 2, 4, 6,12,10, 8)( 5,11)( 7, 9)$
$ 6, 6 $ $18$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4,10,12, 6, 8)$
$ 6, 6 $ $12$ $6$ $( 1, 3, 9, 7, 5,11)( 2, 4,10, 8, 6,12)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 3)( 2,12)( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 4, 4, 4 $ $27$ $4$ $( 1, 4, 3, 2)( 5, 8, 7, 6)( 9,12,11,10)$
$ 12 $ $18$ $12$ $( 1, 4, 7,10, 5,12,11, 6, 9, 8, 3, 2)$
$ 4, 4, 4 $ $9$ $4$ $( 1, 4,11, 2)( 3,10, 5,12)( 6, 9, 8, 7)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$

Group invariants

Order:  $216=2^{3} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [216, 156]
Character table:   
      2  3  3  .  1  .  3   2  3  3  1  2  1  3  3   2  3  1  2
      3  3  .  3  3  3  .   1  1  1  2  1  2  2  .   1  1  3  3

        1a 2a 3a 3b 3c 4a 12a 4b 2b 6a 6b 6c 2c 4c 12b 4d 3d 3e
     2P 1a 1a 3a 3b 3c 2b  6b 2b 1a 3b 3e 3d 1a 2b  6b 2b 3d 3e
     3P 1a 2a 1a 1a 1a 4c  4d 4d 2b 2c 2b 2c 2c 4a  4b 4b 1a 1a
     5P 1a 2a 3a 3b 3c 4a 12a 4b 2b 6a 6b 6c 2c 4c 12b 4d 3d 3e
     7P 1a 2a 3a 3b 3c 4c 12b 4d 2b 6a 6b 6c 2c 4a 12a 4b 3d 3e
    11P 1a 2a 3a 3b 3c 4c 12b 4d 2b 6a 6b 6c 2c 4a 12a 4b 3d 3e

X.1      1  1  1  1  1  1   1  1  1  1  1  1  1  1   1  1  1  1
X.2      1 -1  1  1  1 -1   1  1  1 -1  1 -1 -1 -1   1  1  1  1
X.3      1 -1  1  1  1  1  -1 -1  1 -1  1 -1 -1  1  -1 -1  1  1
X.4      1  1  1  1  1 -1  -1 -1  1  1  1  1  1 -1  -1 -1  1  1
X.5      1 -1  1  1  1  A  -A -A -1  1 -1  1  1 -A   A  A  1  1
X.6      1 -1  1  1  1 -A   A  A -1  1 -1  1  1  A  -A -A  1  1
X.7      1  1  1  1  1  A   A  A -1 -1 -1 -1 -1 -A  -A -A  1  1
X.8      1  1  1  1  1 -A  -A -A -1 -1 -1 -1 -1  A   A  A  1  1
X.9      2  . -1  2 -1  .  -1  2  2  . -1  .  .  .  -1  2  2 -1
X.10     2  . -1  2 -1  .   1 -2  2  . -1  .  .  .   1 -2  2 -1
X.11     2  . -1  2 -1  .   A  B -2  .  1  .  .  .  -A -B  2 -1
X.12     2  . -1  2 -1  .  -A -B -2  .  1  .  .  .   A  B  2 -1
X.13     4  . -2  1  1  .   .  .  . -1  .  2 -4  .   .  . -2  4
X.14     4  . -2  1  1  .   .  .  .  1  . -2  4  .   .  . -2  4
X.15     4  .  1 -2 -2  .   .  .  . -2  .  1  4  .   .  .  1  4
X.16     4  .  1 -2 -2  .   .  .  .  2  . -1 -4  .   .  .  1  4
X.17     8  .  2  2 -1  .   .  .  .  .  .  .  .  .   .  . -4 -4
X.18     8  . -1 -4  2  .   .  .  .  .  .  .  .  .   .  .  2 -4

A = -E(4)
  = -Sqrt(-1) = -i
B = 2*E(4)
  = 2*Sqrt(-1) = 2i