Galois group: 12T272

• Label: 12T272
• Degree: $12$
• Order: $7920$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $M_{11}$

Group invariants

Abstract group:		$M_{11}$
Order:		$7920=2^{4} \cdot 3^{2} \cdot 5 \cdot 11$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$272$
CHM label:		$M_{11}(12)$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,7,3,10,5,9,6,12)(2,11,8,4)$, $(1,6,3,9)(2,7,12,10,4,5,11,8)$

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Low degree siblings

11T6, 22T22

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	$2^{4},1^{4}$	$165$	$2$	$4$	$( 1, 4)( 5, 9)( 6,10)(11,12)$
3A	$3^{3},1^{3}$	$440$	$3$	$6$	$( 1, 8,12)( 3, 7, 9)( 4,10, 5)$
4A	$4^{2},2^{2}$	$990$	$4$	$8$	$( 1, 9, 4, 5)( 2, 7)( 3, 8)( 6,12,10,11)$
5A	$5^{2},1^{2}$	$1584$	$5$	$8$	$( 1, 3,11,12, 2)( 5,10, 8, 9, 6)$
6A	$6,3,2,1$	$1320$	$6$	$8$	$( 1,12, 8)( 2,11)( 3, 5, 7, 4, 9,10)$
8A1	$8,4$	$990$	$8$	$10$	$( 1, 6, 9,12, 4,10, 5,11)( 2, 3, 7, 8)$
8A-1	$8,4$	$990$	$8$	$10$	$( 1,11, 5,10, 4,12, 9, 6)( 2, 8, 7, 3)$
11A1	$11,1$	$720$	$11$	$10$	$( 1, 2, 4, 8,11,12, 7, 3, 6, 5,10)$
11A-1	$11,1$	$720$	$11$	$10$	$( 1,10, 5, 6, 3, 7,12,11, 8, 4, 2)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	3A	4A	5A	6A	8A1	8A-1	11A1	11A-1
	Size	1	165	440	990	1584	1320	990	990	720	720
	2 P	1A	1A	3A	2A	5A	3A	4A	4A	11A-1	11A1
	3 P	1A	2A	1A	4A	5A	2A	8A1	8A-1	11A1	11A-1
	5 P	1A	2A	3A	4A	1A	6A	8A-1	8A1	11A1	11A-1
	11 P	1A	2A	3A	4A	5A	6A	8A1	8A-1	1A	1A
	Type
7920.a.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
7920.a.10a	R	$10$	$2$	$1$	$2$	$0$	$-1$	$0$	$0$	$-1$	$-1$
7920.a.10b1	C	$10$	$-2$	$1$	$0$	$0$	$1$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$	$-1$	$-1$
7920.a.10b2	C	$10$	$-2$	$1$	$0$	$0$	$1$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$	$-1$	$-1$
7920.a.11a	R	$11$	$3$	$2$	$-1$	$1$	$0$	$-1$	$-1$	$0$	$0$
7920.a.16a1	C	$16$	$0$	$-2$	$0$	$1$	$0$	$0$	$0$	$-{\zeta }_{11}^{-2}-1-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$	${\zeta }_{11}^{-2}+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$
7920.a.16a2	C	$16$	$0$	$-2$	$0$	$1$	$0$	$0$	$0$	${\zeta }_{11}^{-2}+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$	$-{\zeta }_{11}^{-2}-1-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$
7920.a.44a	R	$44$	$4$	$-1$	$0$	$-1$	$1$	$0$	$0$	$0$	$0$
7920.a.45a	R	$45$	$-3$	$0$	$1$	$0$	$0$	$-1$	$-1$	$1$	$1$
7920.a.55a	R	$55$	$-1$	$1$	$-1$	$0$	$-1$	$1$	$1$	$0$	$0$

Regular extensions

Data not computed