# Properties

 Label 24T7 Degree $24$ Order $24$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $\SL(2,3)$

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$12$:  $A_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: $A_4$

Degree 6: $A_4$

Degree 8: $\SL(2,3)$

Degree 12: $A_4$

## Low degree siblings

8T12

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

## Group invariants

 Order: $24=2^{3} \cdot 3$ Cyclic: no Abelian: no Solvable: yes GAP id: [24, 3]
 Character table:  2 3 3 1 1 1 1 2 3 1 1 1 1 1 1 . 1a 2a 3a 6a 3b 6b 4a 2P 1a 1a 3b 3b 3a 3a 2a 3P 1a 2a 1a 2a 1a 2a 4a 5P 1a 2a 3b 6b 3a 6a 4a X.1 1 1 1 1 1 1 1 X.2 1 1 A A /A /A 1 X.3 1 1 /A /A A A 1 X.4 2 -2 -1 1 -1 1 . X.5 2 -2 -/A /A -A A . X.6 2 -2 -A A -/A /A . X.7 3 3 . . . . -1 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3