# Properties

 Label 24T31 Degree $24$ Order $48$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_{24}:C_2$

## Group action invariants

 Degree $n$: $24$ Transitive number $t$: $31$ Group: $C_{24}:C_2$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $4$ Generators: (1,24,22,19,18,16,13,12,9,7,5,3,2,23,21,20,17,15,14,11,10,8,6,4), (1,17)(2,18)(3,4)(5,14)(6,13)(7,23)(8,24)(11,20)(12,19)(15,16)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$16$:  $C_8:C_2$
$24$:  $S_3 \times C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $D_{6}$

Degree 8: $C_8:C_2$

Degree 12: $S_3 \times C_4$

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

## Group invariants

 Order: $48=2^{4} \cdot 3$ Cyclic: no Abelian: no Solvable: yes GAP id: [48, 5]
 Character table:  2 4 3 4 3 3 3 3 3 3 3 3 3 3 4 4 3 3 3 3 1 . 1 1 . 1 1 . 1 1 . 1 1 1 1 1 1 1 1a 2a 2b 24a 8a 24b 12a 4a 12b 8b 8c 3a 6a 4b 4c 24c 24d 8d 2P 1a 1a 1a 12b 4c 12b 6a 2b 6a 4c 4b 3a 3a 2b 2b 12a 12a 4b 3P 1a 2a 2b 8b 8c 8b 4b 4a 4c 8d 8a 1a 2b 4c 4b 8d 8d 8b 5P 1a 2a 2b 24a 8a 24b 12a 4a 12b 8b 8c 3a 6a 4b 4c 24c 24d 8d 7P 1a 2a 2b 24d 8c 24c 12b 4a 12a 8d 8a 3a 6a 4c 4b 24b 24a 8b 11P 1a 2a 2b 24d 8c 24c 12b 4a 12a 8d 8a 3a 6a 4c 4b 24b 24a 8b 13P 1a 2a 2b 24b 8a 24a 12a 4a 12b 8b 8c 3a 6a 4b 4c 24d 24c 8d 17P 1a 2a 2b 24b 8a 24a 12a 4a 12b 8b 8c 3a 6a 4b 4c 24d 24c 8d 19P 1a 2a 2b 24c 8c 24d 12b 4a 12a 8d 8a 3a 6a 4c 4b 24a 24b 8b 23P 1a 2a 2b 24c 8c 24d 12b 4a 12a 8d 8a 3a 6a 4c 4b 24a 24b 8b X.1 1 1 1 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 1 -1 -1 -1 X.3 1 -1 1 1 -1 1 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 1 1 1 -1 -1 -1 X.5 1 -1 1 A -A A -1 1 -1 -A A 1 1 -1 -1 -A -A A X.6 1 -1 1 -A A -A -1 1 -1 A -A 1 1 -1 -1 A A -A X.7 1 1 1 A A A -1 -1 -1 -A -A 1 1 -1 -1 -A -A A X.8 1 1 1 -A -A -A -1 -1 -1 A A 1 1 -1 -1 A A -A X.9 2 . 2 -1 . -1 -1 . -1 2 . -1 -1 2 2 -1 -1 2 X.10 2 . 2 1 . 1 -1 . -1 -2 . -1 -1 2 2 1 1 -2 X.11 2 . -2 . . . C . -C . . 2 -2 -C C . . . X.12 2 . -2 . . . -C . C . . 2 -2 C -C . . . X.13 2 . -2 B . -B A . -A . . -1 1 C -C /B -/B . X.14 2 . -2 -/B . /B -A . A . . -1 1 -C C -B B . X.15 2 . -2 /B . -/B -A . A . . -1 1 -C C B -B . X.16 2 . -2 -B . B A . -A . . -1 1 C -C -/B /B . X.17 2 . 2 A . A 1 . 1 C . -1 -1 -2 -2 -A -A -C X.18 2 . 2 -A . -A 1 . 1 -C . -1 -1 -2 -2 A A C A = -E(4) = -Sqrt(-1) = -i B = -E(24)+E(24)^17 C = -2*E(4) = -2*Sqrt(-1) = -2i