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Properties

Label 12T78
Order \(144\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times S_3\wr C_2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $78$
Group :  $C_2\times S_3\wr C_2$
CHM label :  $[2]F_{36}:2_{2}{S_{3}^{2}}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,3)(6,11)(7,10), (1,3,5,7,9,11)(2,4,6,8,10,12), (1,3,5,11,9,7)(2,4,10,8,6,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$
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

12T77 x 4, 12T78 x 3, 18T63 x 4, 24T261 x 4, 24T262 x 4, 24T263 x 2, 24T264 x 2, 36T157 x 2, 36T159 x 4, 36T165 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 $ $6$ $2$ $( 2, 3)( 6,11)( 7,10)$
$ 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 $ $12$ $6$ $( 2, 7,10, 3, 6,11)( 4, 8)( 5, 9)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,12)( 4, 7)( 5, 6)( 8,11)( 9,10)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 4, 4, 4 $ $18$ $4$ $( 1, 2, 4, 7)( 3, 5, 6,12)( 8,11, 9,10)$
$ 4, 4, 4 $ $18$ $4$ $( 1, 2, 4,11)( 3, 5,10,12)( 6, 8, 7, 9)$
$ 6, 6 $ $12$ $6$ $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$
$ 6, 6 $ $12$ $6$ $( 1, 2, 5,10, 9, 6)( 3, 4,11, 8, 7,12)$
$ 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, 186]
Character table:    
      2  4  4  3  3  2  2  3  3  3  3  2  2  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 2d 2e 4a 4b 6b 6c 2f 6d 6e 6f 3b 2g
     2P 1a 1a 1a 1a 3a 3a 1a 1a 2f 2f 3b 3b 1a 3a 3a 3b 3b 1a
     3P 1a 2a 2b 2c 1a 2c 2d 2e 4a 4b 2d 2e 2f 2g 2b 2g 1a 2g
     5P 1a 2a 2b 2c 3a 6a 2d 2e 4a 4b 6b 6c 2f 6d 6e 6f 3b 2g

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  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1
X.6      1  1 -1 -1  1 -1 -1 -1  1  1 -1 -1  1  1 -1  1  1  1
X.7      1  1 -1 -1  1 -1  1  1 -1 -1  1  1  1  1 -1  1  1  1
X.8      1  1  1  1  1  1 -1 -1 -1 -1 -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  . -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  . -2 -2  .  .  1  1  . -2  .  1  1  4
X.14     4  .  .  . -2  . -2  2  .  .  1 -1  .  2  . -1  1 -4
X.15     4  .  .  . -2  .  2 -2  .  . -1  1  .  2  . -1  1 -4
X.16     4  .  .  . -2  .  2  2  .  . -1 -1  . -2  .  1  1  4
X.17     4  .  2 -2  1  1  .  .  .  .  .  .  . -1 -1  2 -2 -4
X.18     4  .  2  2  1 -1  .  .  .  .  .  .  .  1 -1 -2 -2  4