# Properties

 Label 24T21 Degree $24$ Order $48$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $\SL(2,3):C_2$

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$, $A_4\times C_2$

Degree 8: None

Degree 12: $A_4\times C_2$

## Low degree siblings

16T60

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

## Group invariants

 Order: $48=2^{4} \cdot 3$ Cyclic: no Abelian: no Solvable: yes GAP id: [48, 33]
 Character table:  2 4 3 4 2 2 2 2 2 2 2 2 3 4 4 3 1 . 1 1 1 1 1 1 1 1 1 . 1 1 1a 2a 2b 12a 3a 12b 6a 12c 3b 12d 6b 4a 4b 4c 2P 1a 1a 1a 6b 3b 6b 3b 6a 3a 6a 3a 2b 2b 2b 3P 1a 2a 2b 4b 1a 4c 2b 4b 1a 4c 2b 4a 4c 4b 5P 1a 2a 2b 12c 3b 12d 6b 12a 3a 12b 6a 4a 4b 4c 7P 1a 2a 2b 12b 3a 12a 6a 12d 3b 12c 6b 4a 4c 4b 11P 1a 2a 2b 12d 3b 12c 6b 12b 3a 12a 6a 4a 4c 4b X.1 1 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 -1 -1 X.3 1 -1 1 A -A A -A /A -/A /A -/A 1 -1 -1 X.4 1 -1 1 /A -/A /A -/A A -A A -A 1 -1 -1 X.5 1 1 1 -/A -/A -/A -/A -A -A -A -A 1 1 1 X.6 1 1 1 -A -A -A -A -/A -/A -/A -/A 1 1 1 X.7 2 . -2 B -1 -B 1 B -1 -B 1 . D -D X.8 2 . -2 -B -1 B 1 -B -1 B 1 . -D D X.9 2 . -2 C A -C -A -/C /A /C -/A . D -D X.10 2 . -2 -/C /A /C -/A C A -C -A . D -D X.11 2 . -2 /C /A -/C -/A -C A C -A . -D D X.12 2 . -2 -C A C -A /C /A -/C -/A . -D D X.13 3 -1 3 . . . . . . . . -1 3 3 X.14 3 1 3 . . . . . . . . -1 -3 -3 A = -E(3) = (1-Sqrt(-3))/2 = -b3 B = -E(4) = -Sqrt(-1) = -i C = -E(12)^7 D = -2*E(4) = -2*Sqrt(-1) = -2i