# Properties

 Label 24T13 Order $$24$$ n $$24$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_{12}$

## Group action invariants

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

## 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:  $D_{6}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$, $D_{4}$ x 2

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

Degree 8: $D_4$

Degree 12: $D_6$, $D_{12}$ x 2

## Low degree siblings

12T12 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $6$ $2$ $( 1, 2)( 3,23)( 4,24)( 5,22)( 6,21)( 7,20)( 8,19)( 9,18)(10,17)(11,16)(12,15) (13,14)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $6$ $2$ $( 1, 3)( 2, 4)( 5,23)( 6,24)( 7,22)( 8,21)( 9,20)(10,19)(11,18)(12,17)(13,16) (14,15)$ $12, 12$ $2$ $12$ $( 1, 4, 6, 8,10,12,14,16,18,20,22,23)( 2, 3, 5, 7, 9,11,13,15,17,19,21,24)$ $6, 6, 6, 6$ $2$ $6$ $( 1, 6,10,14,18,22)( 2, 5, 9,13,17,21)( 3, 7,11,15,19,24)( 4, 8,12,16,20,23)$ $4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1, 8,14,20)( 2, 7,13,19)( 3, 9,15,21)( 4,10,16,22)( 5,11,17,24)( 6,12,18,23)$ $3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,10,18)( 2, 9,17)( 3,11,19)( 4,12,20)( 5,13,21)( 6,14,22)( 7,15,24) ( 8,16,23)$ $12, 12$ $2$ $12$ $( 1,12,22, 8,18, 4,14,23,10,20, 6,16)( 2,11,21, 7,17, 3,13,24, 9,19, 5,15)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,21)(10,22)(11,24) (12,23)$

## Group invariants

 Order: $24=2^{3} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes GAP id: [24, 6]
 Character table:  2 3 2 2 2 2 2 2 2 3 3 1 . . 1 1 1 1 1 1 1a 2a 2b 12a 6a 4a 3a 12b 2c 2P 1a 1a 1a 6a 3a 2c 3a 6a 1a 3P 1a 2a 2b 4a 2c 4a 1a 4a 2c 5P 1a 2a 2b 12b 6a 4a 3a 12a 2c 7P 1a 2a 2b 12b 6a 4a 3a 12a 2c 11P 1a 2a 2b 12a 6a 4a 3a 12b 2c X.1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 1 1 X.3 1 -1 1 -1 1 -1 1 -1 1 X.4 1 1 -1 -1 1 -1 1 -1 1 X.5 2 . . . -2 . 2 . -2 X.6 2 . . -1 -1 2 -1 -1 2 X.7 2 . . 1 -1 -2 -1 1 2 X.8 2 . . A 1 . -1 -A -2 X.9 2 . . -A 1 . -1 A -2 A = -E(12)^7+E(12)^11 = Sqrt(3) = r3