# Properties

 Label 24T10 Order $$24$$ n $$24$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $S_4$

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $S_4$

Degree 6: $S_3$, $S_4$, $S_4$

Degree 8: $S_4$

Degree 12: $S_4$, $S_4$

## Low degree siblings

4T5, 6T7, 6T8, 8T14, 12T8, 12T9

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

## Group invariants

 Order: $24=2^{3} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes GAP id: [24, 12]
 Character table:  2 3 2 . 2 3 3 1 . 1 . . 1a 2a 3a 4a 2b 2P 1a 1a 3a 2b 1a 3P 1a 2a 1a 4a 2b X.1 1 1 1 1 1 X.2 1 -1 1 -1 1 X.3 2 . -1 . 2 X.4 3 -1 . 1 -1 X.5 3 1 . -1 -1