Properties

Label 12T39
Order \(72\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2.S_3^2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $39$
Group :  $C_2.S_3^2$
CHM label :  $[3^{2}:2]4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (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 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$ x 2
8:  $C_4\times C_2$
12:  $D_{6}$ x 2
24:  $S_3 \times C_4$ x 2
36:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: $S_3^2$

Low degree siblings

12T39, 24T75, 36T32 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 $ $9$ $2$ $( 3,11)( 4,12)( 5, 9)( 6,10)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 4, 8,12)$
$ 12 $ $6$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$
$ 12 $ $6$ $12$ $( 1, 2, 3,12, 5,10, 7, 8, 9, 6,11, 4)$
$ 4, 4, 4 $ $3$ $4$ $( 1, 2, 7, 8)( 3, 4, 9,10)( 5, 6,11,12)$
$ 4, 4, 4 $ $3$ $4$ $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$
$ 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, 2, 2, 2 $ $4$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 8)( 4,10)( 6,12)$
$ 6, 6 $ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2,12,10, 8, 6, 4)$
$ 12 $ $6$ $12$ $( 1, 4,11, 6, 9, 8, 7,10, 5,12, 3, 2)$
$ 12 $ $6$ $12$ $( 1, 4, 3, 6, 5, 8, 7,10, 9,12,11, 2)$
$ 4, 4, 4 $ $3$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
$ 4, 4, 4 $ $3$ $4$ $( 1, 4, 7,10)( 2, 9, 8, 3)( 5,12,11, 6)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 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:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 21]
Character table:   
      2  3  3  1   2   2  3  3  3  2  1  2   2   2  3  3  2  2  3
      3  2  .  2   1   1  1  1  .  2  2  2   1   1  1  1  2  2  2

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

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

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