Galois group: 22T14

• Label: 22T14
• Degree: $22$
• Order: $1320$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $\PGL(2,11)$

Group invariants

Abstract group:		$\PGL(2,11)$
Order:		$1320=2^{3} \cdot 3 \cdot 5 \cdot 11$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: None

Low degree siblings

12T218, 24T2949

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	$2^{8},1^{6}$	$55$	$2$	$8$	$( 1, 7)( 2,11)( 3, 8)( 5, 9)(12,22)(14,21)(15,18)(19,20)$
2B	$2^{11}$	$66$	$2$	$11$	$( 1,20)( 2,17)( 3,15)( 4,16)( 5,14)( 6,19)( 7,18)( 8,13)( 9,22)(10,21)(11,12)$
3A	$3^{6},1^{4}$	$110$	$3$	$12$	$( 1, 2, 5)( 4,10, 6)( 7,11, 9)(12,21,15)(13,17,16)(14,18,22)$
4A	$4^{4},2^{3}$	$110$	$4$	$15$	$( 1,22, 7,12)( 2,14,11,21)( 3,20, 8,19)( 4,17)( 5,18, 9,15)( 6,13)(10,16)$
5A1	$5^{4},1^{2}$	$132$	$5$	$16$	$( 1, 2, 7,10, 3)( 4, 6, 5,11, 8)(12,13,16,19,14)(15,20,17,18,21)$
5A2	$5^{4},1^{2}$	$132$	$5$	$16$	$( 1, 7, 3, 2,10)( 4, 5, 8, 6,11)(12,16,14,13,19)(15,17,21,20,18)$
6A	$6^{2},3^{2},2^{2}$	$110$	$6$	$16$	$( 1, 9, 2, 7, 5,11)( 3, 8)( 4, 6,10)(12,18,21,22,15,14)(13,16,17)(19,20)$
10A1	$10^{2},2$	$132$	$10$	$19$	$( 1,21, 2,15, 7,20,10,17, 3,18)( 4,12, 6,13, 5,16,11,19, 8,14)( 9,22)$
10A3	$10^{2},2$	$132$	$10$	$19$	$( 1,15,10,18, 2,20, 3,21, 7,17)( 4,13,11,14, 6,16, 8,12, 5,19)( 9,22)$
11A	$11^{2}$	$120$	$11$	$20$	$( 1,11, 2,10, 7, 5, 6, 4, 9, 8, 3)(12,14,19,20,15,17,16,18,21,13,22)$
12A1	$12,6,4$	$110$	$12$	$19$	$( 1,21, 9,22, 2,15, 7,14, 5,12,11,18)( 3,19, 8,20)( 4,16, 6,17,10,13)$
12A5	$12,6,4$	$110$	$12$	$19$	$( 1,15,11,22, 5,21, 7,18, 2,12, 9,14)( 3,19, 8,20)( 4,13,10,17, 6,16)$

Malle's constant $a(G)$:     $1/8$

Character table

		1A	2A	2B	3A	4A	5A1	5A2	6A	10A1	10A3	11A	12A1	12A5
	Size	1	55	66	110	110	132	132	110	132	132	120	110	110
	2 P	1A	1A	1A	3A	2A	5A2	5A1	3A	5A1	5A2	11A	6A	6A
	3 P	1A	2A	2B	1A	4A	5A2	5A1	2A	10A3	10A1	11A	4A	4A
	5 P	1A	2A	2B	3A	4A	1A	1A	6A	2B	2B	11A	12A5	12A1
	11 P	1A	2A	2B	3A	4A	5A1	5A2	6A	10A1	10A3	1A	12A1	12A5
	Type
1320.133.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1320.133.1b	R	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$
1320.133.10a	R	$10$	$-2$	$0$	$1$	$2$	$0$	$0$	$1$	$0$	$0$	$-1$	$-1$	$-1$
1320.133.10b	R	$10$	$2$	$0$	$-2$	$0$	$0$	$0$	$2$	$0$	$0$	$-1$	$0$	$0$
1320.133.10c	R	$10$	$-2$	$0$	$1$	$-2$	$0$	$0$	$1$	$0$	$0$	$-1$	$1$	$1$
1320.133.10d1	R	$10$	$2$	$0$	$1$	$0$	$0$	$0$	$-1$	$0$	$0$	$-1$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$	${\zeta }_{12}^{-1}+{\zeta }_{12}$
1320.133.10d2	R	$10$	$2$	$0$	$1$	$0$	$0$	$0$	$-1$	$0$	$0$	$-1$	${\zeta }_{12}^{-1}+{\zeta }_{12}$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$
1320.133.11a	R	$11$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$0$	$-1$	$-1$
1320.133.11b	R	$11$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$0$	$1$	$1$
1320.133.12a1	R	$12$	$0$	$2$	$0$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	$1$	$0$	$0$
1320.133.12a2	R	$12$	$0$	$2$	$0$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	$1$	$0$	$0$
1320.133.12b1	R	$12$	$0$	$-2$	$0$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$1$	$0$	$0$
1320.133.12b2	R	$12$	$0$	$-2$	$0$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$	$1$	$0$	$0$

Regular extensions

Data not computed