# Properties

 Label 12T23 Degree $12$ Order $48$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2 \times S_4$

# Related objects

## Group action invariants

 Degree $n$: $12$ Transitive number $t$: $23$ Group: $C_2 \times S_4$ CHM label: $S_{4}(6d)[x]2=[1/8.2^{6}]S(3)$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $4$ Generators: (2,8)(3,9)(4,10)(5,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11), (2,10)(3,11)(4,8)(5,9)

## 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, 12T24 x 2, 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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1, 1, 1$ $6$ $2$ $( 2, 4)( 3, 5)( 8,10)( 9,11)$ $2, 2, 2, 2, 1, 1, 1, 1$ $3$ $2$ $( 2, 8)( 3, 9)( 4,10)( 5,11)$ $2, 2, 2, 2, 2, 2$ $6$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6, 9)( 7, 8)(10,11)$ $6, 6$ $8$ $6$ $( 1, 2, 5,12, 3, 4)( 6, 9,10, 7, 8,11)$ $4, 4, 2, 2$ $6$ $4$ $( 1, 2, 7, 8)( 3, 6, 9,12)( 4,11)( 5,10)$ $3, 3, 3, 3$ $8$ $3$ $( 1, 3, 5)( 2, 4,12)( 6, 8,10)( 7, 9,11)$ $4, 4, 2, 2$ $6$ $4$ $( 1, 3, 7, 9)( 2, 6, 8,12)( 4,10)( 5,11)$ $2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 6)( 2, 3)( 4,11)( 5,10)( 7,12)( 8, 9)$ $2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$

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