Galois group: 12T37

• Label: 12T37
• Degree: $12$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $S_3\times D_6$

Group invariants

Abstract group:		$S_3\times D_6$
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 $t$:		$37$
CHM label:		$[3^{2}:2]E(4)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,10)(2,5)(3,12)(4,7)(6,9)(8,11)$, $(1,5)(2,10)(4,8)(7,11)$, $(2,6,10)(4,8,12)$, $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_2^2$ x 7
$6$:	$S_3$ x 2
$8$:	$C_2^3$
$12$:	$D_{6}$ x 6
$24$:	$S_3 \times C_2^2$ x 2
$36$:	$S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $S_3^2$

Low degree siblings

12T37, 18T29 x 4, 24T73, 36T34 x 2, 36T40 x 4

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

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	6A	6B	6C	6D	6E	6F	6G
	Size	1	1	3	3	3	3	9	9	2	2	4	2	2	4	6	6	6	6
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	3A	3B	3C	3A	3B	3C	3B	3B	3A	3A
	3 P	1A	2A	2B	2C	2D	2E	2F	2G	1A	1A	1A	2A	2A	2A	2B	2C	2D	2E
	Type
72.46.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.46.1b	R	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$
72.46.1c	R	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$
72.46.1d	R	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$
72.46.1e	R	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$
72.46.1f	R	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
72.46.1g	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$
72.46.1h	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
72.46.2a	R	$2$	$2$	$0$	$0$	$2$	$2$	$0$	$0$	$-1$	$2$	$-1$	$-1$	$2$	$-1$	$0$	$0$	$-1$	$-1$
72.46.2b	R	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$2$	$-1$	$-1$	$2$	$-1$	$-1$	$-1$	$-1$	$0$	$0$
72.46.2c	R	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$2$	$-1$	$-1$	$-2$	$1$	$1$	$1$	$-1$	$0$	$0$
72.46.2d	R	$2$	$-2$	$0$	$0$	$-2$	$2$	$0$	$0$	$-1$	$2$	$-1$	$1$	$-2$	$1$	$0$	$0$	$1$	$-1$
72.46.2e	R	$2$	$-2$	$0$	$0$	$2$	$-2$	$0$	$0$	$-1$	$2$	$-1$	$1$	$-2$	$1$	$0$	$0$	$-1$	$1$
72.46.2f	R	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$2$	$-1$	$-1$	$-2$	$1$	$1$	$-1$	$1$	$0$	$0$
72.46.2g	R	$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$2$	$-1$	$-1$	$2$	$-1$	$-1$	$1$	$1$	$0$	$0$
72.46.2h	R	$2$	$2$	$0$	$0$	$-2$	$-2$	$0$	$0$	$-1$	$2$	$-1$	$-1$	$2$	$-1$	$0$	$0$	$1$	$1$
72.46.4a	R	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$-2$	$1$	$-2$	$-2$	$1$	$0$	$0$	$0$	$0$
72.46.4b	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$-2$	$1$	$2$	$2$	$-1$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed