# Properties

 Label 24T11 Degree $24$ Order $24$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2^2\times S_3$

## Group action invariants

 Degree $n$: $24$ Transitive number $t$: $11$ Group: $C_2^2\times S_3$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $24$ Generators: (1,8)(2,7)(3,21)(4,22)(5,11)(6,12)(9,15)(10,16)(13,19)(14,20)(17,24)(18,23), (1,7)(2,8)(3,5)(4,6)(9,24)(10,23)(11,21)(12,22)(13,20)(14,19)(15,17)(16,18), (1,6)(2,5)(3,15)(4,16)(7,11)(8,12)(9,21)(10,22)(13,17)(14,18)(19,24)(20,23)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 3: $S_3$

Degree 4: $C_2^2$ x 7

Degree 6: $S_3$, $D_{6}$ x 6

Degree 8: $C_2^3$

Degree 12: $D_6$ x 3, $S_3 \times C_2^2$ x 4

## Low degree siblings

12T10 x 4

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 2)( 3,11)( 4,12)( 5,21)( 6,22)( 7, 8)( 9,17)(10,18)(13,14)(15,24)(16,23) (19,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 3)( 2, 4)( 5,24)( 6,23)( 7,22)( 8,21)( 9,20)(10,19)(11,17)(12,18)(13,16) (14,15)$ $6, 6, 6, 6$ $2$ $6$ $( 1, 4,18,19, 9,11)( 2, 3,17,20,10,12)( 5, 8,22,23,13,15)( 6, 7,21,24,14,16)$ $6, 6, 6, 6$ $2$ $6$ $( 1, 5, 9,13,18,22)( 2, 6,10,14,17,21)( 3, 7,12,16,20,24)( 4, 8,11,15,19,23)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,11)( 8,12)( 9,21)(10,22)(13,17)(14,18)(19,24) (20,23)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,24)(10,23)(11,21)(12,22)(13,20)(14,19)(15,17) (16,18)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 8)( 2, 7)( 3,21)( 4,22)( 5,11)( 6,12)( 9,15)(10,16)(13,19)(14,20)(17,24) (18,23)$ $3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 9,18)( 2,10,17)( 3,12,20)( 4,11,19)( 5,13,22)( 6,14,21)( 7,16,24) ( 8,15,23)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,22)(10,21)(11,23) (12,24)$ $6, 6, 6, 6$ $2$ $6$ $( 1,15,18, 8, 9,23)( 2,16,17, 7,10,24)( 3, 6,20,21,12,14)( 4, 5,19,22,11,13)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,19)( 2,20)( 3,10)( 4, 9)( 5,23)( 6,24)( 7,14)( 8,13)(11,18)(12,17)(15,22) (16,21)$

## Group invariants

 Order: $24=2^{3} \cdot 3$ Cyclic: no Abelian: no Solvable: yes GAP id: [24, 14]
 Character table:  2 3 3 3 2 2 3 3 3 2 3 2 3 3 1 . . 1 1 . . 1 1 1 1 1 1a 2a 2b 6a 6b 2c 2d 2e 3a 2f 6c 2g 2P 1a 1a 1a 3a 3a 1a 1a 1a 3a 1a 3a 1a 3P 1a 2a 2b 2g 2f 2c 2d 2e 1a 2f 2e 2g 5P 1a 2a 2b 6a 6b 2c 2d 2e 3a 2f 6c 2g X.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 X.3 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 X.5 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 X.6 1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 X.7 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 X.8 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 X.9 2 . . -1 -1 . . 2 -1 2 -1 2 X.10 2 . . -1 1 . . -2 -1 -2 1 2 X.11 2 . . 1 -1 . . -2 -1 2 1 -2 X.12 2 . . 1 1 . . 2 -1 -2 -1 -2