# Properties

 Label 24T5 Degree $24$ Order $24$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3:Q_8$

## Group action invariants

 Degree $n$: $24$ Transitive number $t$: $5$ Group: $C_3:Q_8$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $24$ Generators: (1,20,2,19)(3,17,4,18)(5,15,6,16)(7,13,8,14)(9,12,10,11)(21,23,22,24), (1,7,18,23,9,15,2,8,17,24,10,16)(3,13,20,5,11,21,4,14,19,6,12,22)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $Q_8$
$12$:  $D_{6}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

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

Degree 8: $Q_8$

Degree 12: $D_6$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $24=2^{3} \cdot 3$ Cyclic: no Abelian: no Solvable: yes GAP id: [24, 4]
 Character table:  2 3 3 2 2 2 2 2 2 2 3 1 1 . . 1 1 1 1 1 1a 2a 4a 4b 12a 12b 3a 6a 4c 2P 1a 1a 2a 2a 6a 6a 3a 3a 2a 3P 1a 2a 4a 4b 4c 4c 1a 2a 4c 5P 1a 2a 4a 4b 12b 12a 3a 6a 4c 7P 1a 2a 4a 4b 12b 12a 3a 6a 4c 11P 1a 2a 4a 4b 12a 12b 3a 6a 4c 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 -2 . . . . 2 -2 . X.6 2 2 . . -1 -1 -1 -1 2 X.7 2 2 . . 1 1 -1 -1 -2 X.8 2 -2 . . A -A -1 1 . X.9 2 -2 . . -A A -1 1 . A = -E(12)^7+E(12)^11 = Sqrt(3) = r3