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Properties

Label 12T103
Order \(192\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_2^2:S_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $103$
Group :  $C_2\times C_2^2:S_4$
CHM label :  $1/2[E(4)^{3}]S(3)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11)(2,4)(3,9)(5,7)(6,12)(8,10), (3,12)(6,9), (1,7)(3,9)(4,10)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $4$

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:  $S_4$ x 3
48:  $S_4\times C_2$ x 3
96:  $V_4^2:S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$, $S_4\times C_2$ x 2

Low degree siblings

12T100 x 3, 12T101 x 6, 12T103 x 5, 12T106, 16T429, 24T432 x 3, 24T485 x 3, 24T486 x 6, 24T487 x 6, 24T488 x 3, 24T489 x 3, 24T490 x 3, 24T491 x 2, 24T492 x 6, 24T493 x 6, 24T508 x 3, 24T509 x 6, 24T510, 24T511, 32T2212 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3,12)( 6, 9)$
$ 4, 4, 1, 1, 1, 1 $ $12$ $4$ $( 2, 3, 8, 9)( 5,12,11, 6)$
$ 4, 4, 1, 1, 1, 1 $ $12$ $4$ $( 2, 3,11, 6)( 5,12, 8, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 5)( 3,12)( 6, 9)( 8,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 2, 8)( 3, 6)( 5,11)( 9,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3, 9)( 5,11)( 6,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2,11)( 3, 6)( 5, 8)( 9,12)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 6)( 4,11)( 5,10)( 7, 8)( 9,12)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 9)( 4,11)( 5,10)( 6,12)( 7, 8)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 2, 3)( 4,11, 6)( 5,12,10)( 7, 8, 9)$
$ 6, 6 $ $32$ $6$ $( 1, 2, 3,10, 5,12)( 4,11, 6, 7, 8, 9)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2, 4,11)( 3,12)( 5, 7, 8,10)( 6, 9)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2, 7, 8)( 3,12)( 4,11,10, 5)( 6, 9)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2,10, 5)( 3, 6)( 4,11, 7, 8)( 9,12)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2,10, 5)( 3, 9)( 4,11, 7, 8)( 6,12)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 4)( 2, 5)( 3, 9)( 6,12)( 7,10)( 8,11)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 7)( 2, 5)( 3, 9)( 4,10)( 6,12)( 8,11)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)$

Group invariants

Order:  $192=2^{6} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [192, 1538]
Character table:    
      2  6  6  4  4  6  5  6  6  4  4  1  1  4  4  4  4  6  5  6  6
      3  1  .  .  .  .  .  .  .  .  .  1  1  .  .  .  .  .  .  .  1

        1a 2a 4a 4b 2b 2c 2d 2e 2f 2g 3a 6a 4c 4d 4e 4f 2h 2i 2j 2k
     2P 1a 1a 2d 2e 1a 1a 1a 1a 1a 1a 3a 3a 2e 2d 2b 2b 1a 1a 1a 1a
     3P 1a 2a 4a 4b 2b 2c 2d 2e 2f 2g 1a 2k 4c 4d 4e 4f 2h 2i 2j 2k
     5P 1a 2a 4a 4b 2b 2c 2d 2e 2f 2g 3a 6a 4c 4d 4e 4f 2h 2i 2j 2k

X.1      1  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 -1 -1
X.3      1 -1  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  1  1
X.5      2 -2  .  .  2 -2  2  2  .  . -1  1  .  .  .  . -2  2 -2 -2
X.6      2  2  .  .  2  2  2  2  .  . -1 -1  .  .  .  .  2  2  2  2
X.7      3 -3 -1  1  3  1 -1 -1 -1  1  .  . -1  1  1 -1  1 -1  1 -3
X.8      3 -3  1 -1  3  1 -1 -1  1 -1  .  .  1 -1 -1  1  1 -1  1 -3
X.9      3 -1 -1  1 -1 -1  3 -1 -1 -1  .  .  1 -1  1  1  3 -1 -1  3
X.10     3 -1  1 -1 -1 -1  3 -1  1  1  .  . -1  1 -1 -1  3 -1 -1  3
X.11     3  1 -1 -1 -1  1  3 -1  1 -1  .  .  1  1  1 -1 -3 -1  1 -3
X.12     3  1  1  1 -1  1  3 -1 -1  1  .  . -1 -1 -1  1 -3 -1  1 -3
X.13     3  3 -1 -1  3 -1 -1 -1  1  1  .  . -1 -1  1  1 -1 -1 -1  3
X.14     3  3  1  1  3 -1 -1 -1 -1 -1  .  .  1  1 -1 -1 -1 -1 -1  3
X.15     3 -1 -1  1 -1 -1 -1  3  1  1  .  .  1 -1 -1 -1 -1 -1  3  3
X.16     3 -1  1 -1 -1 -1 -1  3 -1 -1  .  . -1  1  1  1 -1 -1  3  3
X.17     3  1 -1 -1 -1  1 -1  3 -1  1  .  .  1  1 -1  1  1 -1 -3 -3
X.18     3  1  1  1 -1  1 -1  3  1 -1  .  . -1 -1  1 -1  1 -1 -3 -3
X.19     6  2  .  . -2 -2 -2 -2  .  .  .  .  .  .  .  .  2  2  2 -6
X.20     6 -2  .  . -2  2 -2 -2  .  .  .  .  .  .  .  . -2  2 -2  6