# Properties

 Label 24T14 Degree $24$ Order $24$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3:D_4$

## Group action invariants

 Degree $n$: $24$ Transitive number $t$: $14$ Group: $C_3:D_4$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $24$ Generators: (1,9)(2,10)(3,8)(4,7)(5,6)(11,23)(12,24)(13,22)(14,21)(15,20)(16,19)(17,18), (1,19,14,7)(2,20,13,8)(3,6,15,18)(4,5,16,17)(9,11,21,24)(10,12,22,23)

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 6: $S_3$, $D_{6}$ x 2

Degree 8: $D_4$

Degree 12: $D_6$, $(C_6\times C_2):C_2$, $(C_6\times C_2):C_2$

## Low degree siblings

12T13, 12T15

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

## Group invariants

 Order: $24=2^{3} \cdot 3$ Cyclic: no Abelian: no Solvable: yes GAP id: [24, 8]
 Character table:  2 3 2 2 2 2 2 2 2 3 3 1 . . 1 1 1 1 1 1 1a 2a 4a 6a 6b 2b 3a 6c 2c 2P 1a 1a 2c 3a 3a 1a 3a 3a 1a 3P 1a 2a 4a 2b 2c 2b 1a 2b 2c 5P 1a 2a 4a 6c 6b 2b 3a 6a 2c X.1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 1 1 X.3 1 -1 1 -1 1 -1 1 -1 1 X.4 1 1 -1 -1 1 -1 1 -1 1 X.5 2 . . 1 -1 -2 -1 1 2 X.6 2 . . -1 -1 2 -1 -1 2 X.7 2 . . . -2 . 2 . -2 X.8 2 . . A 1 . -1 -A -2 X.9 2 . . -A 1 . -1 A -2 A = -E(3)+E(3)^2 = -Sqrt(-3) = -i3