Galois group: 12T8

• Label: 12T8
• Degree: $12$
• Order: $24$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $S_4$

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$8$
Group:		$S_4$
CHM label:		$S_{4}(12d)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,2)(3,5)(4,6)(7,9)(8,10), (1,3,6,12)(2,4,7,10)(5,8,11,9)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$6$:	$S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: $S_4$

Degree 6: $S_4$

Low degree siblings

4T5, 6T7, 6T8, 8T14, 12T9, 24T10

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

Group invariants

Order:		$24=2^{3} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		24.12
Character table:

		1A	2A	2B	3A	4A
	Size	1	3	6	8	6
	2 P	1A	1A	1A	3A	2A
	3 P	1A	2A	2B	1A	4A
	Type
24.12.1a	R	$1$	$1$	$1$	$1$	$1$
24.12.1b	R	$1$	$1$	$-1$	$1$	$-1$
24.12.2a	R	$2$	$2$	$0$	$-1$	$0$
24.12.3a	R	$3$	$-1$	$1$	$0$	$-1$
24.12.3b	R	$3$	$-1$	$-1$	$0$	$1$