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Properties

Label 12T80
Order \(144\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3^2:C_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $80$
Group :  $S_3^2:C_4$
CHM label :  $[2]F_{36}:2_{2}{3^{2}:4}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,3)(6,11)(7,10), (2,6,10)(3,7,11), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11), (1,3)(2,12)(4,6,8,10)(5,7,9,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $D_{4}$ x 2, $C_4\times C_2$
16:  $C_2^2:C_4$
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: $C_3^2:D_4$

Low degree siblings

12T79 x 2, 12T80, 24T213, 24T215, 24T265 x 2, 24T266 x 2, 24T267, 24T268, 36T156 x 2, 36T158, 36T164

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

Group invariants

Order:  $144=2^{4} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [144, 115]
Character table:    
      2  4  4  3  3  2  2  3  3   2   2  3  3  4  2  2  2  2  4
      3  2  .  1  1  2  1  .  .   1   1  1  1  .  2  1  2  2  2

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

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

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