Galois group: 12T12

• Label: 12T12
• Degree: $12$
• Order: $24$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $D_{12}$

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$12$
Group:		$D_{12}$
CHM label:		$1/2[3:2]cD(4)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,11)(2,10)(3,9)(4,8)(5,7), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)

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: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Low degree siblings

12T12, 24T13

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 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 1, 1 $	$6$	$2$	$( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)$
	$ 2, 2, 2, 2, 2, 2 $	$6$	$2$	$( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
	$ 12 $	$2$	$12$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$
	$ 6, 6 $	$2$	$6$	$( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
	$ 4, 4, 4 $	$2$	$4$	$( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
	$ 3, 3, 3, 3 $	$2$	$3$	$( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
	$ 12 $	$2$	$12$	$( 1, 6,11, 4, 9, 2, 7,12, 5,10, 3, 8)$
	$ 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$

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	$2$	$-2$	$0$	$0$	$-1$	$0$	$1$	${\zeta }_{12}^{-1}+{\zeta }_{12}$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$