Galois group: 12T44

• Label: 12T44
• Degree: $12$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_3:S_4$

Group invariants

Abstract group:		$C_3:S_4$
Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$44$
CHM label:		$1/2[3:2]S(4)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,11)(2,10)(3,9)(4,8)(5,7)$, $(1,7,10)(2,5,11)(3,6,9)$, $(2,8,11)(3,6,12)(4,7,10)$, $(1,5,9)(2,6,10)(3,7,11)(4,8,12)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$6$:	$S_3$ x 4
$18$:	$C_3^2:C_2$
$24$:	$S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: $S_4$

Degree 6: None

Low degree siblings

12T44 x 2, 18T37, 18T40, 24T79 x 3, 36T23, 36T56

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	$2^{6}$	$3$	$2$	$6$	$( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
2B	$2^{5},1^{2}$	$18$	$2$	$5$	$( 1,12)( 3, 7)( 4, 9)( 5, 8)( 6,10)$
3A	$3^{4}$	$2$	$3$	$8$	$( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
3B	$3^{4}$	$8$	$3$	$8$	$( 1,11, 6)( 2, 9, 7)( 3,10, 5)( 4, 8,12)$
3C	$3^{3},1^{3}$	$8$	$3$	$6$	$( 1, 7,10)( 2, 5,11)( 3, 6, 9)$
3D	$3^{4}$	$8$	$3$	$8$	$( 1, 3, 2)( 4,12, 8)( 5, 7, 6)( 9,11,10)$
4A	$4^{3}$	$18$	$4$	$9$	$( 1, 7, 4,10)( 2, 9,11,12)( 3, 8, 6, 5)$
6A	$6^{2}$	$6$	$6$	$10$	$( 1,12, 5, 4, 9, 8)( 2, 7, 6,11,10, 3)$

Malle's constant $a(G)$:     $1/5$

Character table

		1A	2A	2B	3A	3B	3C	3D	4A	6A
	Size	1	3	18	2	8	8	8	18	6
	2 P	1A	1A	1A	3A	3B	3C	3D	2A	3A
	3 P	1A	2A	2B	1A	1A	1A	1A	4A	2A
	Type
72.43.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.43.1b	R	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$
72.43.2a	R	$2$	$2$	$0$	$-1$	$-1$	$-1$	$2$	$0$	$-1$
72.43.2b	R	$2$	$2$	$0$	$-1$	$-1$	$2$	$-1$	$0$	$-1$
72.43.2c	R	$2$	$2$	$0$	$-1$	$2$	$-1$	$-1$	$0$	$-1$
72.43.2d	R	$2$	$2$	$0$	$2$	$-1$	$-1$	$-1$	$0$	$2$
72.43.3a	R	$3$	$-1$	$1$	$3$	$0$	$0$	$0$	$-1$	$-1$
72.43.3b	R	$3$	$-1$	$-1$	$3$	$0$	$0$	$0$	$1$	$-1$
72.43.6a	R	$6$	$-2$	$0$	$-3$	$0$	$0$	$0$	$0$	$1$

Regular extensions

Data not computed