Galois group: 22T26

• Label: 22T26
• Degree: $22$
• Order: $15840$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $C_2\times M_{11}$

Group action invariants

Degree $n$:		$22$
Transitive number $t$:		$26$
Group:		$C_2\times M_{11}$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,21,14,15,10,8,12,3)(2,22,13,16,9,7,11,4)(5,20,6,19)(17,18), (1,3,5,11,18,2,4,6,12,17)(7,16,20,22,14,8,15,19,21,13)(9,10)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$7920$:	$M_{11}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 11: $M_{11}$

Low degree siblings

22T27, 24T12204, 44T140

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

Group invariants

Order:		$15840=2^{5} \cdot 3^{2} \cdot 5 \cdot 11$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		15840.q
Character table:

	Size
	2 P
	3 P
	5 P
	11 P
	Type