# Properties

 Label 24T12 Degree $24$ Order $24$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_4\times S_3$

## Group action invariants

 Degree $n$: $24$ Transitive number $t$: $12$ Group: $C_4\times S_3$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $24$ Generators: (1,6)(2,5)(3,15)(4,16)(7,12)(8,11)(9,21)(10,22)(13,17)(14,18)(19,23)(20,24), (1,16,5,20,9,23,13,3,18,7,22,11)(2,15,6,19,10,24,14,4,17,8,21,12)

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_4$ x 2, $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2

Degree 8: $C_4\times C_2$

Degree 12: $D_6$, $S_3 \times C_4$ x 2

## Low degree siblings

12T11 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,21)( 6,22)( 7, 8)( 9,17)(10,18)(13,14)(15,23)(16,24) (19,20)$ $12, 12$ $2$ $12$ $( 1, 3, 5, 7, 9,11,13,16,18,20,22,23)( 2, 4, 6, 8,10,12,14,15,17,19,21,24)$ $4, 4, 4, 4, 4, 4$ $3$ $4$ $( 1, 4,13,15)( 2, 3,14,16)( 5,24,18,12)( 6,23,17,11)( 7,10,20,21)( 8, 9,19,22)$ $6, 6, 6, 6$ $2$ $6$ $( 1, 5, 9,13,18,22)( 2, 6,10,14,17,21)( 3, 7,11,16,20,23)( 4, 8,12,15,19,24)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,12)( 8,11)( 9,21)(10,22)(13,17)(14,18)(19,23) (20,24)$ $4, 4, 4, 4, 4, 4$ $1$ $4$ $( 1, 7,13,20)( 2, 8,14,19)( 3, 9,16,22)( 4,10,15,21)( 5,11,18,23)( 6,12,17,24)$ $4, 4, 4, 4, 4, 4$ $3$ $4$ $( 1, 8,13,19)( 2, 7,14,20)( 3,17,16, 6)( 4,18,15, 5)( 9,24,22,12)(10,23,21,11)$ $3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1, 9,18)( 2,10,17)( 3,11,20)( 4,12,19)( 5,13,22)( 6,14,21)( 7,16,23) ( 8,15,24)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,22)(10,21)(11,23) (12,24)$ $12, 12$ $2$ $12$ $( 1,16, 5,20, 9,23,13, 3,18, 7,22,11)( 2,15, 6,19,10,24,14, 4,17, 8,21,12)$ $4, 4, 4, 4, 4, 4$ $1$ $4$ $( 1,20,13, 7)( 2,19,14, 8)( 3,22,16, 9)( 4,21,15,10)( 5,23,18,11)( 6,24,17,12)$

## Group invariants

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