Galois group: 24T13

• Label: 24T13
• Degree: $24$
• Order: $24$
• 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
$\card{\Aut(F/K)}$:		$24$
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)

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

Label	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
Nilpotency class:		not nilpotent
Label:		24.6
Character table:

		1A	2A	2B	2C	3A	4A	6A	12A1	12A5
	Size	1	1	6	6	2	2	2	2	2
	2 P	1A	1A	1A	1A	3A	2A	3A	6A	6A
	3 P	1A	2A	2B	2C	1A	4A	2A	4A	4A
	Type
24.6.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
24.6.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$
24.6.1c	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
24.6.1d	R	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$
24.6.2a	R	$2$	$2$	$0$	$0$	$-1$	$2$	$-1$	$-1$	$-1$
24.6.2b	R	$2$	$-2$	$0$	$0$	$2$	$0$	$-2$	$0$	$0$
24.6.2c	R	$2$	$2$	$0$	$0$	$-1$	$-2$	$-1$	$1$	$1$
24.6.2d1	R	$2$	$-2$	$0$	$0$	$-1$	$0$	$1$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$	${\zeta }_{12}^{-1}+{\zeta }_{12}$
24.6.2d2	R	$2$	$-2$	$0$	$0$	$-1$	$0$	$1$	${\zeta }_{12}^{-1}+{\zeta }_{12}$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$