Galois group: 12T36

• Label: 12T36
• Degree: $12$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $\SOPlus(4,2)$

Group invariants

Abstract group:		$\SOPlus(4,2)$
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 $t$:		$36$
CHM label:		$F_{36}:2(12d)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,3,5,7,9,11)(2,4,6,8,10,12)$, $(1,9)(2,3)(6,7)(8,12)(10,11)$

Low degree resolvents

$\card{(G/N)}$	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: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: $C_3^2:D_4$

Low degree siblings

6T13 x 2, 9T16, 12T34 x 2, 12T35 x 2, 12T36, 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	Index	Representative
1A	$1^{12}$	$1$	$1$	$0$	$()$
2A	$2^{5},1^{2}$	$6$	$2$	$5$	$( 1,12)( 4, 5)( 6,10)( 7,11)( 8, 9)$
2B	$2^{6}$	$6$	$2$	$6$	$( 1, 6)( 2, 5)( 3, 4)( 7,12)( 8,11)( 9,10)$
2C	$2^{6}$	$9$	$2$	$6$	$( 1, 8)( 2,11)( 3,10)( 4, 5)( 6, 7)( 9,12)$
3A	$3^{2},1^{6}$	$4$	$3$	$4$	$( 1, 9, 5)( 4,12, 8)$
3B	$3^{4}$	$4$	$3$	$8$	$( 1, 9, 5)( 2, 6,10)( 3, 7,11)( 4,12, 8)$
4A	$4^{3}$	$18$	$4$	$9$	$( 1,11, 8, 2)( 3,12,10, 9)( 4, 6, 5, 7)$
6A	$6,2^{2},1^{2}$	$12$	$6$	$7$	$( 1, 4, 9,12, 5, 8)( 6,10)( 7,11)$
6B	$6^{2}$	$12$	$6$	$10$	$( 1, 2, 9, 6, 5,10)( 3, 8, 7, 4,11,12)$

Malle's constant $a(G)$:     $1/4$

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	$4$	$0$	$-2$	$0$	$-2$	$1$	$0$	$0$	$1$

Regular extensions

Data not computed