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 action invariants

Degree $n$:		$12$
Transitive number $t$:		$16$
Group:		$S_3^2$
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

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

Group invariants

Order:		$36=2^{2} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		36.10
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	$4$	$0$	$0$	$0$	$-2$	$-2$	$1$	$0$	$0$