Galois group: 12T16

• Label: 12T16
• Degree: $12$
• Order: $36$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $S_3^2$

Group invariants

Abstract group:		$S_3^2$
Order:		$36=2^{2} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$16$
CHM label:		$[3^{2}]E(4)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$6$
Generators:		$(1,10)(2,5)(3,12)(4,7)(6,9)(8,11)$, $(2,10,6)(4,8,12)$, $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$

Low degree resolvents

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

6T9, 9T8, 18T9, 18T11 x 2, 36T13

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

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	3A	3B	3C	6A	6B
	Size	1	3	3	9	2	2	4	6	6
	2 P	1A	1A	1A	1A	3A	3B	3C	3A	3B
	3 P	1A	2A	2B	2C	1A	1A	1A	2A	2B
	Type
36.10.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
36.10.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
36.10.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$
36.10.1d	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$
36.10.2a	R	$2$	$0$	$2$	$0$	$2$	$-1$	$-1$	$0$	$-1$
36.10.2b	R	$2$	$2$	$0$	$0$	$-1$	$2$	$-1$	$-1$	$0$
36.10.2c	R	$2$	$-2$	$0$	$0$	$-1$	$2$	$-1$	$1$	$0$
36.10.2d	R	$2$	$0$	$-2$	$0$	$2$	$-1$	$-1$	$0$	$1$
36.10.4a	R	$4$	$0$	$0$	$0$	$-2$	$-2$	$1$	$0$	$0$

Regular extensions

Data not computed