# Properties

 Label 24T28 Degree $24$ Order $48$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_6.C_2^2$

## Group action invariants

 Degree $n$: $24$ Transitive number $t$: $28$ Group: $D_6.C_2^2$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $4$ Generators: (1,24,10,8,17,15,2,23,9,7,18,16)(3,13,12,22,19,6,4,14,11,21,20,5), (1,24,2,23)(3,22,4,21)(5,20,6,19)(7,18,8,17)(9,15,10,16)(11,14,12,13), (1,20,2,19)(3,9,4,10)(5,24,6,23)(7,13,8,14)(11,17,12,18)(15,21,16,22)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3
$16$:  $Q_8:C_2$
$24$:  $S_3 \times C_2^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $D_{6}$ x 3

Degree 8: $Q_8:C_2$

Degree 12: $S_3 \times C_2^2$

## Low degree siblings

24T28 x 2

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

## Group invariants

 Order: $48=2^{4} \cdot 3$ Cyclic: no Abelian: no Solvable: yes GAP id: [48, 41]
 Character table:  2 4 3 4 3 2 2 3 4 3 4 3 3 3 2 3 3 1 . 1 . 1 1 . . 1 . 1 1 1 1 1 1a 2a 2b 2c 12a 12b 2d 4a 4b 4c 3a 6a 4d 12c 4e 2P 1a 1a 1a 1a 6a 6a 1a 2b 2b 2b 3a 3a 2b 6a 2b 3P 1a 2a 2b 2c 4e 4d 2d 4c 4b 4a 1a 2b 4d 4b 4e 5P 1a 2a 2b 2c 12a 12b 2d 4a 4b 4c 3a 6a 4d 12c 4e 7P 1a 2a 2b 2c 12a 12b 2d 4c 4b 4a 3a 6a 4d 12c 4e 11P 1a 2a 2b 2c 12a 12b 2d 4c 4b 4a 3a 6a 4d 12c 4e X.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 X.3 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 X.5 1 -1 1 1 -1 1 -1 1 -1 1 1 1 1 -1 -1 X.6 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 1 -1 X.7 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 X.8 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 X.9 2 . 2 . -1 -1 . . 2 . -1 -1 2 -1 2 X.10 2 . 2 . -1 1 . . -2 . -1 -1 -2 1 2 X.11 2 . 2 . 1 -1 . . -2 . -1 -1 2 1 -2 X.12 2 . 2 . 1 1 . . 2 . -1 -1 -2 -1 -2 X.13 2 . -2 . . . . A . -A 2 -2 . . . X.14 2 . -2 . . . . -A . A 2 -2 . . . X.15 4 . -4 . . . . . . . -2 2 . . . A = -2*E(4) = -2*Sqrt(-1) = -2i