Galois group: 22T22

• Label: 22T22
• Degree: $22$
• Order: $7920$
• Cyclic: no
• Abelian: no
• Solvable: no
• Transitivity: $1$
• Primitive: no
• $p$-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 $n$:		$22$
Transitive number $t$:		$22$
Parity:		$1$
Transitivity:		1
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,11,3,19,14)(2,12,4,20,13)(5,17,16,9,7)(6,18,15,10,8)$, $(1,22,15,4,20)(2,21,16,3,19)(5,12,14,9,8)(6,11,13,10,7)$

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 11: $M_{11}$

Low degree siblings

11T6, 12T272

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	$1^{22}$	$1$	$1$	$0$	$()$
2A	$2^{8},1^{6}$	$165$	$2$	$8$	$( 1,22)( 2,21)( 3,16)( 4,15)( 5,14)( 6,13)(11,17)(12,18)$
3A	$3^{6},1^{4}$	$440$	$3$	$12$	$( 1, 5, 8)( 2, 6, 7)( 3,11,21)( 4,12,22)( 9,15,20)(10,16,19)$
4A	$4^{4},2^{2},1^{2}$	$990$	$4$	$14$	$( 1,11,22,17)( 2,12,21,18)( 3, 6,16,13)( 4, 5,15,14)( 9,10)(19,20)$
5A	$5^{4},1^{2}$	$1584$	$5$	$16$	$( 1, 6,16,18,13)( 2, 5,15,17,14)( 3, 7,20,12,22)( 4, 8,19,11,21)$
6A	$6^{2},3^{2},2^{2}$	$1320$	$6$	$16$	$( 1, 3, 5,11, 8,21)( 2, 4, 6,12, 7,22)( 9,20,15)(10,19,16)(13,17)(14,18)$
8A1	$8^{2},4,2$	$990$	$8$	$18$	$( 1, 5,11,15,22,14,17, 4)( 2, 6,12,16,21,13,18, 3)( 7, 8)( 9,19,10,20)$
8A-1	$8^{2},4,2$	$990$	$8$	$18$	$( 1, 4,17,14,22,15,11, 5)( 2, 3,18,13,21,16,12, 6)( 7, 8)( 9,20,10,19)$
11A1	$11^{2}$	$720$	$11$	$20$	$( 1, 6,20,21,17,14,16,10, 3,12, 8)( 2, 5,19,22,18,13,15, 9, 4,11, 7)$
11A-1	$11^{2}$	$720$	$11$	$20$	$( 1, 8,12, 3,10,16,14,17,21,20, 6)( 2, 7,11, 4, 9,15,13,18,22,19, 5)$

Malle's constant $a(G)$:     $1/8$

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