Galois group: 12T34

• Label: 12T34
• Degree: $12$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $\SOPlus(4,2)$

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$34$
Group:		$\SOPlus(4,2)$
CHM label:		$F_{36}:2(12e)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,8)(2,3)(4,5)(6,7)(9,12)(10,11), (1,3,5,7,9,11)(2,4,6,8,10,12)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $C_3^2:D_4$

Low degree siblings

6T13 x 2, 9T16, 12T34, 12T35 x 2, 12T36 x 2, 18T34 x 2, 18T36, 24T72 x 2, 36T53, 36T54 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	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 1, 1, 1, 1 $	$9$	$2$	$( 4, 8)( 5, 9)( 6,10)( 7,11)$
	$ 3, 3, 1, 1, 1, 1, 1, 1 $	$4$	$3$	$( 2, 6,10)( 3, 7,11)$
	$ 4, 4, 2, 2 $	$18$	$4$	$( 1, 2)( 3,12)( 4, 7, 8,11)( 5, 6, 9,10)$
	$ 6, 6 $	$12$	$6$	$( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
	$ 2, 2, 2, 2, 2, 2 $	$6$	$2$	$( 1, 3)( 2,12)( 4, 6)( 5, 7)( 8,10)( 9,11)$
	$ 2, 2, 2, 2, 2, 2 $	$6$	$2$	$( 1, 4)( 2, 3)( 5,12)( 6, 7)( 8, 9)(10,11)$
	$ 6, 2, 2, 2 $	$12$	$6$	$( 1, 4, 9,12, 5, 8)( 2, 3)( 6,11)( 7,10)$
	$ 3, 3, 3, 3 $	$4$	$3$	$( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$

Group invariants

Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		72.40
Character table:

		1A	2A	2B	2C	3A	3B	4A	6A	6B
	Size	1	6	6	9	4	4	18	12	12
	2 P	1A	1A	1A	1A	3A	3B	2C	3A	3B
	3 P	1A	2A	2B	2C	1A	1A	4A	2A	2B
	Type
72.40.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.40.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
72.40.1c	R	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$
72.40.1d	R	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$
72.40.2a	R	$2$	$0$	$0$	$-2$	$2$	$2$	$0$	$0$	$0$
72.40.4a	R	$4$	$0$	$2$	$0$	$-2$	$1$	$0$	$0$	$-1$
72.40.4b	R	$4$	$2$	$0$	$0$	$1$	$-2$	$0$	$-1$	$0$
72.40.4c	R	$4$	$-2$	$0$	$0$	$1$	$-2$	$0$	$1$	$0$
72.40.4d	R	$4$	$0$	$-2$	$0$	$-2$	$1$	$0$	$0$	$1$