Galois group: 22T22

• Label: 22T22
• Degree: $22$
• Order: $7920$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $M_{11}$

Group action invariants

Degree $n$:		$22$
Transitive number $t$:		$22$
Group:		$M_{11}$
Parity:		$1$
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	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$165$	$2$	$( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,14)(10,13)(11,22)(12,21)$
	$ 4, 4, 4, 4, 2, 2, 1, 1 $	$990$	$4$	$( 1,22, 7,11)( 2,21, 8,12)( 3,13, 5,10)( 4,14, 6, 9)(15,16)(19,20)$
	$ 8, 8, 4, 2 $	$990$	$8$	$( 1,13,22, 5, 7,10,11, 3)( 2,14,21, 6, 8, 9,12, 4)(15,20,16,19)(17,18)$
	$ 8, 8, 4, 2 $	$990$	$8$	$( 1, 3,11,10, 7, 5,22,13)( 2, 4,12, 9, 8, 6,21,14)(15,19,16,20)(17,18)$
	$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $	$440$	$3$	$( 1,11,13)( 2,12,14)( 7,22,10)( 8,21, 9)(15,17,20)(16,18,19)$
	$ 6, 6, 3, 3, 2, 2 $	$1320$	$6$	$( 1,10,11, 7,13,22)( 2, 9,12, 8,14,21)( 3, 5)( 4, 6)(15,20,17)(16,19,18)$
	$ 11, 11 $	$720$	$11$	$( 1,21, 3,14,17, 8,12,20,16,10, 6)( 2,22, 4,13,18, 7,11,19,15, 9, 5)$
	$ 11, 11 $	$720$	$11$	$( 1, 6,10,16,20,12, 8,17,14, 3,21)( 2, 5, 9,15,19,11, 7,18,13, 4,22)$
	$ 5, 5, 5, 5, 1, 1 $	$1584$	$5$	$( 3,20,16,10, 8)( 4,19,15, 9, 7)( 5,22,11,13,18)( 6,21,12,14,17)$

Group invariants

Order:		$7920=2^{4} \cdot 3^{2} \cdot 5 \cdot 11$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		7920.a
Character table:

	Size
	2 P
	3 P
	5 P
	11 P
	Type