Galois group: 12T24

• Label: 12T24
• Degree: $12$
• Order: $48$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_2 \times S_4$

Group invariants

Abstract group:		$C_2 \times S_4$
Order:		$48=2^{4} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$24$
CHM label:		$S_{4}(6c)[x]2$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(2,8)(3,9)(4,10)(5,11)$, $(1,7)(2,10)(3,11)(4,8)(5,9)(6,12)$, $(1,5,9)(2,6,10)(3,7,11)(4,8,12)$, $(1,12)(2,3)(4,5)(6,7)(8,9)(10,11)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$6$:	$S_3$
$12$:	$D_{6}$
$24$:	$S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$, $S_4$, $S_4\times C_2$

Low degree siblings

6T11 x 2, 8T24 x 2, 12T21, 12T22, 12T23 x 2, 12T24, 16T61, 24T46, 24T47, 24T48 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	Index	Representative
1A	$1^{12}$	$1$	$1$	$0$	$()$
2A	$2^{6}$	$1$	$2$	$6$	$( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
2B	$2^{6}$	$3$	$2$	$6$	$( 1, 6)( 2, 3)( 4,11)( 5,10)( 7,12)( 8, 9)$
2C	$2^{4},1^{4}$	$3$	$2$	$4$	$( 1, 7)( 2, 8)( 3, 9)( 6,12)$
2D	$2^{6}$	$6$	$2$	$6$	$( 1, 8)( 2, 7)( 3, 6)( 4,11)( 5,10)( 9,12)$
2E	$2^{6}$	$6$	$2$	$6$	$( 1, 3)( 2,12)( 4,10)( 5,11)( 6, 8)( 7, 9)$
3A	$3^{4}$	$8$	$3$	$8$	$( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
4A	$4^{2},1^{4}$	$6$	$4$	$6$	$( 1, 3, 7, 9)( 2, 6, 8,12)$
4B	$4^{2},2^{2}$	$6$	$4$	$8$	$( 1, 8, 7, 2)( 3,12, 9, 6)( 4, 5)(10,11)$
6A	$6^{2}$	$8$	$6$	$10$	$( 1, 4, 9,12, 5, 8)( 2, 7,10, 3, 6,11)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	2D	2E	3A	4A	4B	6A
	Size	1	1	3	3	6	6	8	6	6	8
	2 P	1A	1A	1A	1A	1A	1A	3A	2C	2C	3A
	3 P	1A	2A	2B	2C	2D	2E	1A	4A	4B	2A
	Type
48.48.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.48.1b	R	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
48.48.1c	R	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
48.48.1d	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$
48.48.2a	R	$2$	$2$	$2$	$2$	$0$	$0$	$-1$	$0$	$0$	$-1$
48.48.2b	R	$2$	$-2$	$-2$	$2$	$0$	$0$	$-1$	$0$	$0$	$1$
48.48.3a	R	$3$	$3$	$-1$	$-1$	$1$	$1$	$0$	$-1$	$-1$	$0$
48.48.3b	R	$3$	$3$	$-1$	$-1$	$-1$	$-1$	$0$	$1$	$1$	$0$
48.48.3c	R	$3$	$-3$	$1$	$-1$	$-1$	$1$	$0$	$-1$	$1$	$0$
48.48.3d	R	$3$	$-3$	$1$	$-1$	$1$	$-1$	$0$	$1$	$-1$	$0$

Regular extensions

Data not computed