Properties

Label 12T24
Order \(48\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2 \times S_4$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $24$
Group :  $C_2 \times S_4$
CHM label :  $S_{4}(6c)[x]2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,8)(3,9)(4,10)(5,11), (1,7)(2,10)(3,11)(4,8)(5,9)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11)
$|\Aut(F/K)|$:  $4$

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:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$, $S_4$, $S_4\times C_2$

Low degree siblings

6T11 x 2, 8T24 x 2, 12T21, 12T22, 12T23 x 2, 12T24, 16T61, 24T46, 24T47, 24T48 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 $ $1$ $1$ $()$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 4, 8,10)( 3, 5, 9,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3, 9)( 4,10)( 5,11)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 6, 6 $ $8$ $6$ $( 1, 2, 5,12, 3, 4)( 6, 9,10, 7, 8,11)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 2, 7, 8)( 3, 6, 9,12)( 4, 5)(10,11)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 3, 5)( 2, 4,12)( 6, 8,10)( 7, 9,11)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 3)( 2,12)( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 6)( 2, 3)( 4,11)( 5,10)( 7,12)( 8, 9)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 48]
Character table:   
      2  4  3  4  3  1  3  1  3  4  4
      3  1  .  .  .  1  .  1  .  .  1

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

X.1      1  1  1  1  1  1  1  1  1  1
X.2      1 -1  1 -1  1 -1  1 -1  1  1
X.3      1 -1  1  1 -1  1  1 -1 -1 -1
X.4      1  1  1 -1 -1 -1  1  1 -1 -1
X.5      2  .  2  . -1  . -1  .  2  2
X.6      2  .  2  .  1  . -1  . -2 -2
X.7      3 -1 -1 -1  .  1  .  1  1 -3
X.8      3 -1 -1  1  . -1  .  1 -1  3
X.9      3  1 -1 -1  .  1  . -1 -1  3
X.10     3  1 -1  1  . -1  . -1  1 -3