# Properties

 Label 24T47 Degree $24$ Order $48$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2\times S_4$

## Group action invariants

 Degree $n$: $24$ Transitive number $t$: $47$ Group: $C_2\times S_4$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $8$ Generators: (1,6,10,13,18,22)(2,5,9,14,17,21)(3,7,12,15,19,24)(4,8,11,16,20,23), (1,23,14,12)(2,24,13,11)(3,10)(4,9)(5,8,18,19)(6,7,17,20)(15,22)(16,21)

## 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: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2, $S_4$, $S_4$, $S_4\times C_2$ x 2

Degree 8: None

Degree 12: $D_6$, $S_4$, $C_2\times S_4$, $C_2 \times S_4$ x 2, $C_2 \times S_4$ x 2

## Low degree siblings

6T11 x 2, 8T24 x 2, 12T21, 12T22, 12T23 x 2, 12T24 x 2, 16T61, 24T46, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $2$ $( 3,16)( 4,15)( 5,18)( 6,17)( 9,22)(10,21)(11,24)(12,23)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 2)( 3, 4)( 5,17)( 6,18)( 7, 8)( 9,10)(11,23)(12,24)(13,14)(15,16)(19,20) (21,22)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $6$ $2$ $( 1, 3)( 2, 4)( 5,12)( 6,11)( 7,22)( 8,21)( 9,20)(10,19)(13,15)(14,16)(17,24) (18,23)$ $4, 4, 4, 4, 2, 2, 2, 2$ $6$ $4$ $( 1, 3,14,16)( 2, 4,13,15)( 5,23)( 6,24)( 7,22,20, 9)( 8,21,19,10)(11,17) (12,18)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $6$ $2$ $( 1, 4)( 2, 3)( 5,24)( 6,23)( 7,21)( 8,22)( 9,19)(10,20)(11,18)(12,17)(13,16) (14,15)$ $4, 4, 4, 4, 2, 2, 2, 2$ $6$ $4$ $( 1, 4,14,15)( 2, 3,13,16)( 5,11)( 6,12)( 7,21,20,10)( 8,22,19, 9)(17,23) (18,24)$ $3, 3, 3, 3, 3, 3, 3, 3$ $8$ $3$ $( 1, 5,21)( 2, 6,22)( 3, 8,12)( 4, 7,11)( 9,13,17)(10,14,18)(15,20,24) (16,19,23)$ $6, 6, 6, 6$ $8$ $6$ $( 1, 6,10,13,18,22)( 2, 5, 9,14,17,21)( 3, 7,12,15,19,24)( 4, 8,11,16,20,23)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,21)(10,22)(11,23) (12,24)$

## Group invariants

 Order: $48=2^{4} \cdot 3$ Cyclic: no Abelian: no Solvable: yes GAP id: [48, 48]
 Character table:  2 4 4 4 3 3 3 3 1 1 4 3 1 . . . . . . 1 1 1 1a 2a 2b 2c 4a 2d 4b 3a 6a 2e 2P 1a 1a 1a 1a 2a 1a 2a 3a 3a 1a 3P 1a 2a 2b 2c 4a 2d 4b 1a 2e 2e 5P 1a 2a 2b 2c 4a 2d 4b 3a 6a 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 -2 . . . . -1 1 -2 X.6 2 2 2 . . . . -1 -1 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