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Properties

Label 12T112
Order \(192\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4^2:C_3:C_2^2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $112$
Group :  $C_4^2:C_3:C_2^2$
CHM label :  $[1/2[1/2.2^{2}]^{3}]2S_{4}(6)_{8}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,3)(6,7)(8,9), (2,3)(4,10)(5,11)(6,7), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,3)(2,12)(4,10)(5,11)(6,8)(7,9)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
24:  $S_4$
48:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4\times C_2$

Low degree siblings

12T112, 12T113 x 2, 12T114 x 2, 12T115 x 2, 16T431, 24T530 x 2, 24T531 x 2, 24T532 x 2, 24T533 x 2, 24T534 x 2, 24T535, 24T536 x 2, 24T537 x 2, 24T538 x 2, 24T539, 24T540, 24T541, 32T2145, 32T2146 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 $ $12$ $2$ $( 4,11)( 5,10)( 6, 7)( 8, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 4, 4, 1, 1, 1, 1 $ $12$ $4$ $( 2, 4, 3, 5)( 8,11, 9,10)$
$ 8, 2, 1, 1 $ $24$ $8$ $( 2, 4, 9,11, 3, 5, 8,10)( 6, 7)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 8, 3, 9)( 4,10, 5,11)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6, 8)( 7, 9)(10,11)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3,12)( 4,10)( 5,11)( 6, 9)( 7, 8)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 2, 4)( 3, 5,12)( 6, 8,11)( 7, 9,10)$
$ 6, 6 $ $32$ $6$ $( 1, 2, 4, 6, 9,11)( 3, 5, 7, 8,10,12)$
$ 8, 4 $ $24$ $8$ $( 1, 2, 6, 8,12, 3, 7, 9)( 4,11, 5,10)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2,12, 3)( 4,11)( 5,10)( 6, 9, 7, 8)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 6,12, 7)( 2, 3)( 4,11, 5,10)( 8, 9)$
$ 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 6)( 2, 8)( 3, 9)( 4,10)( 5,11)( 7,12)$

Group invariants

Order:  $192=2^{6} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [192, 956]
Character table:    
      2  6  4  6  4  3  5  4  4  1  1  3  4  5  4
      3  1  .  .  .  .  .  .  .  1  1  .  .  .  1

        1a 2a 2b 4a 8a 4b 2c 2d 3a 6a 8b 4c 4d 2e
     2P 1a 1a 1a 2b 4b 2b 1a 1a 3a 3a 4d 2b 2b 1a
     3P 1a 2a 2b 4a 8a 4b 2c 2d 1a 2e 8b 4c 4d 2e
     5P 1a 2a 2b 4a 8a 4b 2c 2d 3a 6a 8b 4c 4d 2e
     7P 1a 2a 2b 4a 8a 4b 2c 2d 3a 6a 8b 4c 4d 2e

X.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
X.3      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
X.5      2 -2  2  .  .  2  .  . -1  1  .  .  2 -2
X.6      2  2  2  .  .  2  .  . -1 -1  .  .  2  2
X.7      3 -1  3 -1  1 -1 -1 -1  .  .  1 -1 -1  3
X.8      3 -1  3  1 -1 -1  1  1  .  . -1  1 -1  3
X.9      3  1  3 -1 -1 -1 -1  1  .  .  1  1 -1 -3
X.10     3  1  3  1  1 -1  1 -1  .  . -1 -1 -1 -3
X.11     6  . -2 -2  .  2  2  .  .  .  .  . -2  .
X.12     6  . -2  .  . -2  . -2  .  .  .  2  2  .
X.13     6  . -2  .  . -2  .  2  .  .  . -2  2  .
X.14     6  . -2  2  .  2 -2  .  .  .  .  . -2  .