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Properties

Label 12T82
Order \(144\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:S_3.D_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $82$
Group :  $C_3:S_3.D_4$
CHM label :  $[(1/4.2^{3})^{2}]F_{36}(6)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(4,5)(8,9), (1,5,9)(4,8,12), (2,10)(3,11)(4,8)(5,9), (1,7)(2,8,10,4)(3,9,11,5)(6,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: $C_3^2:C_4$

Low degree siblings

12T82 x 7, 24T272 x 4, 24T273 x 4, 24T274 x 4, 36T126 x 2, 36T141 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 $ $9$ $2$ $( 4, 8)( 5, 9)( 6,10)( 7,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 2, 3)( 6, 7)(10,11)$
$ 2, 2, 2, 2, 2, 1, 1 $ $18$ $2$ $( 2, 3)( 4, 8)( 5, 9)( 6,11)( 7,10)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 3, 7,11)$
$ 6, 1, 1, 1, 1, 1, 1 $ $4$ $6$ $( 2, 7,10, 3, 6,11)$
$ 4, 4, 2, 2 $ $18$ $4$ $( 1, 2)( 3,12)( 4, 7, 8,11)( 5, 6, 9,10)$
$ 4, 4, 2, 2 $ $18$ $4$ $( 1, 2)( 3,12)( 4,11, 8, 7)( 5,10, 9, 6)$
$ 4, 4, 4 $ $18$ $4$ $( 1, 2, 4, 7)( 3, 5, 6,12)( 8,11, 9,10)$
$ 4, 4, 4 $ $18$ $4$ $( 1, 2, 4,11)( 3, 5,10,12)( 6, 8, 7, 9)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)$
$ 6, 2, 2, 2 $ $4$ $6$ $( 1, 4, 9,12, 5, 8)( 2, 3)( 6, 7)(10,11)$
$ 6, 3, 3 $ $4$ $6$ $( 1, 4, 9,12, 5, 8)( 2, 6,10)( 3, 7,11)$
$ 6, 6 $ $4$ $6$ $( 1, 4, 9,12, 5, 8)( 2, 7,10, 3, 6,11)$
$ 6, 3, 3 $ $4$ $6$ $( 1, 4, 9,12, 5, 8)( 2,10, 6)( 3,11, 7)$
$ 3, 3, 2, 2, 2 $ $4$ $6$ $( 1, 5, 9)( 2, 3)( 4, 8,12)( 6, 7)(10,11)$
$ 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, 136]
Character table:    
      2  4  4  3  3  2  2  3  3  3  3  4  2  2  2  2  2  2  4
      3  2  .  2  .  2  2  .  .  .  .  .  2  2  2  2  2  2  2

        1a 2a 2b 2c 3a 6a 4a 4b 4c 4d 2d 6b 6c 6d 6e 6f 3b 2e
     2P 1a 1a 1a 1a 3a 3a 2a 2a 2d 2d 1a 3a 3b 3b 3b 3a 3b 1a
     3P 1a 2a 2b 2c 1a 2b 4b 4a 4d 4c 2d 2e 2b 2e 2b 2b 1a 2e
     5P 1a 2a 2b 2c 3a 6a 4a 4b 4c 4d 2d 6b 6c 6d 6e 6f 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 -1  1 -1  1 -1 -1  1  1
X.6      1 -1 -1  1  1 -1 -A  A  A -A -1  1 -1  1 -1 -1  1  1
X.7      1 -1  1 -1  1  1  A -A  A -A -1  1  1  1  1  1  1  1
X.8      1 -1  1 -1  1  1 -A  A -A  A -1  1  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  . -4  . -2  2  .  .  .  .  . -2 -1  1 -1  2  1  4
X.12     4  . -4  .  1 -1  .  .  .  .  .  1  2 -2  2 -1 -2  4
X.13     4  .  4  . -2 -2  .  .  .  .  . -2  1  1  1 -2  1  4
X.14     4  .  4  .  1  1  .  .  .  .  .  1 -2 -2 -2  1 -2  4
X.15     4  .  .  . -2  .  .  .  .  .  .  2 -3 -1  3  .  1 -4
X.16     4  .  .  . -2  .  .  .  .  .  .  2  3 -1 -3  .  1 -4
X.17     4  .  .  .  1 -3  .  .  .  .  . -1  .  2  .  3 -2 -4
X.18     4  .  .  .  1  3  .  .  .  .  . -1  .  2  . -3 -2 -4

A = -E(4)
  = -Sqrt(-1) = -i