Galois group: 22T38

• Label: 22T38
• Degree: $22$
• Order: $443520$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $M_{22}$

Group invariants

Abstract group:		$M_{22}$
Order:		$443520=2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$22$
Transitive number $t$:		$38$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,17,9,5,10,22,8)(3,20,15,12,19,11,14)(4,21,16,13,7,18,6)$, $(1,5,10)(2,17,12)(3,8,4)(6,16,19)(9,18,21)(14,20,22)$

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 11: None

Low degree siblings

There are no siblings 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	$2^{8},1^{6}$	$1155$	$2$	$8$	$( 1,15)( 2, 4)( 3, 9)( 6,18)( 7,16)(10,20)(13,21)(17,19)$
3A	$3^{6},1^{4}$	$12320$	$3$	$12$	$( 1, 6, 3)( 2,19,10)( 4,17,20)( 5,14,22)( 8,12,11)( 9,15,18)$
4A	$4^{4},2^{2},1^{2}$	$13860$	$4$	$14$	$( 1,13,20,15)( 2,19)( 3,17, 5, 8)( 4,11,22, 6)( 7,21)( 9,12,18,14)$
4B	$4^{4},2^{2},1^{2}$	$27720$	$4$	$14$	$( 1,20)( 2,12, 9,17)( 3,15,14,16)( 5, 8)( 7,18,22,13)(10,11,19,21)$
5A	$5^{4},1^{2}$	$88704$	$5$	$16$	$( 1,15,14, 6,11)( 2,20, 8,13, 9)( 3,22,16,10, 5)( 7,12,21,19,17)$
6A	$6^{2},3^{2},2^{2}$	$36960$	$6$	$16$	$( 1, 9, 6,15, 3,18)( 2,20,19, 4,10,17)( 5,22,14)( 7,16)( 8,11,12)(13,21)$
7A1	$7^{3},1$	$63360$	$7$	$18$	$( 1,18,20, 8,15, 3,21)( 2, 6, 5,10, 7,17, 4)( 9,11,14,13,22,16,19)$
7A-1	$7^{3},1$	$63360$	$7$	$18$	$( 1,21, 3,15, 8,20,18)( 2, 4,17, 7,10, 5, 6)( 9,19,16,22,13,14,11)$
8A	$8^{2},4,2$	$55440$	$8$	$18$	$( 1, 5,13, 8,20, 3,15,17)( 2, 7,19,21)( 4,14,11, 9,22,12, 6,18)(10,16)$
11A1	$11^{2}$	$40320$	$11$	$20$	$( 1, 8,20,14,21,18, 6,15,12, 4,17)( 2,11,19,22, 3,10,13, 5, 9, 7,16)$
11A-1	$11^{2}$	$40320$	$11$	$20$	$( 1,17, 4,12,15, 6,18,21,14,20, 8)( 2,16, 7, 9, 5,13,10, 3,22,19,11)$

Malle's constant $a(G)$:     $1/8$

Character table

		1A	2A	3A	4A	4B	5A	6A	7A1	7A-1	8A	11A1	11A-1
	Size	1	1155	12320	13860	27720	88704	36960	63360	63360	55440	40320	40320
	2 P	1A	1A	3A	2A	2A	5A	3A	7A1	7A-1	4A	11A-1	11A1
	3 P	1A	2A	1A	4A	4B	5A	2A	7A-1	7A1	8A	11A1	11A-1
	5 P	1A	2A	3A	4A	4B	1A	6A	7A-1	7A1	8A	11A1	11A-1
	7 P	1A	2A	3A	4A	4B	5A	6A	1A	1A	8A	11A-1	11A1
	11 P	1A	2A	3A	4A	4B	5A	6A	7A1	7A-1	8A	1A	1A
	Type
443520.a.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
443520.a.21a	R	$21$	$5$	$3$	$1$	$1$	$1$	$-1$	$0$	$0$	$-1$	$-1$	$-1$
443520.a.45a1	C	$45$	$-3$	$0$	$1$	$1$	$0$	$0$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-1$	$1$	$1$
443520.a.45a2	C	$45$	$-3$	$0$	$1$	$1$	$0$	$0$	${\zeta }_7^{-3}+{\zeta }_7+{\zeta }_7^2$	$-{\zeta }_7^{-3}-1-{\zeta }_7-{\zeta }_7^2$	$-1$	$1$	$1$
443520.a.55a	R	$55$	$7$	$1$	$3$	$-1$	$0$	$1$	$-1$	$-1$	$1$	$0$	$0$
443520.a.99a	R	$99$	$3$	$0$	$3$	$-1$	$-1$	$0$	$1$	$1$	$-1$	$0$	$0$
443520.a.154a	R	$154$	$10$	$1$	$-2$	$2$	$-1$	$1$	$0$	$0$	$0$	$0$	$0$
443520.a.210a	R	$210$	$2$	$3$	$-2$	$-2$	$0$	$-1$	$0$	$0$	$0$	$1$	$1$
443520.a.231a	R	$231$	$7$	$-3$	$-1$	$-1$	$1$	$1$	$0$	$0$	$-1$	$0$	$0$
443520.a.280a1	C	$280$	$-8$	$1$	$0$	$0$	$0$	$1$	$0$	$0$	$0$	$-{\zeta }_{11}^{-2}-1-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$	${\zeta }_{11}^{-2}+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$
443520.a.280a2	C	$280$	$-8$	$1$	$0$	$0$	$0$	$1$	$0$	$0$	$0$	${\zeta }_{11}^{-2}+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$	$-{\zeta }_{11}^{-2}-1-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$
443520.a.385a	R	$385$	$1$	$-2$	$1$	$1$	$0$	$-2$	$0$	$0$	$1$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$\left(5 x^{4}+34 x^{3}-119 x^{2}+212 x-164\right)^{4} \left(19 x^{3}-12 x^{2}+28 x+32\right)^{2}-2^{22} \left(x^{2}-x+3\right)^{11}/\left(11 t^{2}+1\right)$