# Properties

 Label 24T42 Degree $24$ Order $48$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3:SD_{16}$

## Group action invariants

 Degree $n$: $24$ Transitive number $t$: $42$ Group: $C_3:SD_{16}$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $6$ Generators: (1,15,2,16)(3,13,4,14)(5,11,6,12)(7,10,8,9)(17,24,18,23)(19,21,20,22), (1,18,9,2,17,10)(3,7,11,15,19,24)(4,8,12,16,20,23)(5,22,14)(6,21,13)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$16$:  $QD_{16}$
$24$:  $(C_6\times C_2):C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $S_3$

Degree 8: $QD_{16}$

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

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

## Group invariants

 Order: $48=2^{4} \cdot 3$ Cyclic: no Abelian: no Solvable: yes GAP id: [48, 16]
 Character table:  2 4 2 4 2 3 3 2 2 3 3 3 2 3 1 1 1 . . . 1 1 1 1 1 1 1a 2a 2b 4a 8a 8b 12a 6a 3a 6b 4b 6c 2P 1a 1a 1a 2b 4b 4b 6b 3a 3a 3a 2b 3a 3P 1a 2a 2b 4a 8a 8b 4b 2a 1a 2b 4b 2a 5P 1a 2a 2b 4a 8b 8a 12a 6c 3a 6b 4b 6a 7P 1a 2a 2b 4a 8b 8a 12a 6a 3a 6b 4b 6c 11P 1a 2a 2b 4a 8a 8b 12a 6c 3a 6b 4b 6a X.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 X.3 1 -1 1 1 -1 -1 1 -1 1 1 1 -1 X.4 1 1 1 -1 -1 -1 1 1 1 1 1 1 X.5 2 . 2 . . . -2 . 2 2 -2 . X.6 2 -2 2 . . . -1 1 -1 -1 2 1 X.7 2 2 2 . . . -1 -1 -1 -1 2 -1 X.8 2 . -2 . A -A . . 2 -2 . . X.9 2 . -2 . -A A . . 2 -2 . . X.10 2 . 2 . . . 1 B -1 -1 -2 -B X.11 2 . 2 . . . 1 -B -1 -1 -2 B X.12 4 . -4 . . . . . -2 2 . . A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 B = -E(3)+E(3)^2 = -Sqrt(-3) = -i3