Galois group: 22T18

• Label: 22T18
• Degree: $22$
• Order: $2420$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{11}^2:C_{20}$

Group invariants

Abstract group:		$C_{11}^2:C_{20}$
Order:		$2420=2^{2} \cdot 5 \cdot 11^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$5$:	$C_5$
$10$:	$C_{10}$
$20$:	20T1

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: None

Low degree siblings

22T18 x 5, 44T62 x 6

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	$2^{10},1^{2}$	$121$	$2$	$10$	$( 1, 5)( 2, 4)( 6,11)( 7,10)( 8, 9)(12,22)(13,21)(14,20)(15,19)(16,18)$
4A1	$4^{5},2$	$121$	$4$	$16$	$( 1,12, 5,22)( 2,20, 4,14)( 3,17)( 6,19,11,15)( 7,16,10,18)( 8,13, 9,21)$
4A-1	$4^{5},2$	$121$	$4$	$16$	$( 1,22, 5,12)( 2,14, 4,20)( 3,17)( 6,15,11,19)( 7,18,10,16)( 8,21, 9,13)$
5A1	$5^{4},1^{2}$	$121$	$5$	$16$	$( 1, 6, 4, 7, 8)( 2,10, 9, 5,11)(12,19,14,16,13)(15,20,18,21,22)$
5A-1	$5^{4},1^{2}$	$121$	$5$	$16$	$( 1, 8, 7, 4, 6)( 2,11, 5, 9,10)(12,13,16,14,19)(15,22,21,18,20)$
5A2	$5^{4},1^{2}$	$121$	$5$	$16$	$( 1, 4, 8, 6, 7)( 2, 9,11,10, 5)(12,14,13,19,16)(15,18,22,20,21)$
5A-2	$5^{4},1^{2}$	$121$	$5$	$16$	$( 1, 7, 6, 8, 4)( 2, 5,10,11, 9)(12,16,19,13,14)(15,21,20,22,18)$
10A1	$10^{2},1^{2}$	$121$	$10$	$18$	$( 1,10, 6, 9, 4, 5, 7,11, 8, 2)(12,18,19,21,14,22,16,15,13,20)$
10A-1	$10^{2},1^{2}$	$121$	$10$	$18$	$( 1, 2, 8,11, 7, 5, 4, 9, 6,10)(12,20,13,15,16,22,14,21,19,18)$
10A3	$10^{2},1^{2}$	$121$	$10$	$18$	$( 1, 9, 7, 2, 6, 5, 8,10, 4,11)(12,21,16,20,19,22,13,18,14,15)$
10A-3	$10^{2},1^{2}$	$121$	$10$	$18$	$( 1,11, 4,10, 8, 5, 6, 2, 7, 9)(12,15,14,18,13,22,19,20,16,21)$
11A	$11^{2}$	$20$	$11$	$20$	$( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,13,14,15,16,17,18,19,20,21,22)$
11B	$11^{2}$	$20$	$11$	$20$	$( 1, 9, 6, 3,11, 8, 5, 2,10, 7, 4)(12,18,13,19,14,20,15,21,16,22,17)$
11C	$11,1^{11}$	$20$	$11$	$10$	$(12,20,17,14,22,19,16,13,21,18,15)$
11D	$11^{2}$	$20$	$11$	$20$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,14,16,18,20,22,13,15,17,19,21)$
11E	$11^{2}$	$20$	$11$	$20$	$( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,21,19,17,15,13,22,20,18,16,14)$
11F	$11^{2}$	$20$	$11$	$20$	$( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,20,17,14,22,19,16,13,21,18,15)$
20A1	$20,2$	$121$	$20$	$20$	$( 1,13,10,20, 6,12, 9,18, 4,19, 5,21, 7,14,11,22, 8,16, 2,15)( 3,17)$
20A-1	$20,2$	$121$	$20$	$20$	$( 1,15, 2,16, 8,22,11,14, 7,21, 5,19, 4,18, 9,12, 6,20,10,13)( 3,17)$
20A3	$20,2$	$121$	$20$	$20$	$( 1,20, 9,19, 7,22, 2,13, 6,18, 5,14, 8,15,10,12, 4,21,11,16)( 3,17)$
20A-3	$20,2$	$121$	$20$	$20$	$( 1,16,11,21, 4,12,10,15, 8,14, 5,18, 6,13, 2,22, 7,19, 9,20)( 3,17)$
20A7	$20,2$	$121$	$20$	$20$	$( 1,18,11,13, 4,22,10,19, 8,20, 5,16, 6,21, 2,12, 7,15, 9,14)( 3,17)$
20A-7	$20,2$	$121$	$20$	$20$	$( 1,14, 9,15, 7,12, 2,21, 6,16, 5,20, 8,19,10,22, 4,13,11,18)( 3,17)$
20A9	$20,2$	$121$	$20$	$20$	$( 1,19, 2,18, 8,12,11,20, 7,13, 5,15, 4,16, 9,22, 6,14,10,21)( 3,17)$
20A-9	$20,2$	$121$	$20$	$20$	$( 1,21,10,14, 6,22, 9,16, 4,15, 5,13, 7,20,11,12, 8,18, 2,19)( 3,17)$

Malle's constant $a(G)$:     $1/10$

Character table

		1A	2A	4A1	4A-1	5A1	5A-1	5A2	5A-2	10A1	10A-1	10A3	10A-3	11A	11B	11C	11D	11E	11F	20A1	20A-1	20A3	20A-3	20A7	20A-7	20A9	20A-9
	Size	1	121	121	121	121	121	121	121	121	121	121	121	20	20	20	20	20	20	121	121	121	121	121	121	121	121
	2 P	1A	1A	2A	2A	5A2	5A-2	5A-1	5A1	5A1	5A-1	5A-2	5A2	11A	11B	11C	11D	11E	11F	10A1	10A-1	10A3	10A-3	10A-3	10A3	10A-1	10A1
	5 P	1A	2A	4A1	4A-1	1A	1A	1A	1A	2A	2A	2A	2A	11A	11B	11C	11D	11E	11F	4A1	4A-1	4A-1	4A1	4A-1	4A1	4A1	4A-1
	11 P	1A	2A	4A-1	4A1	5A1	5A-1	5A2	5A-2	10A1	10A-1	10A3	10A-3	1A	1A	1A	1A	1A	1A	20A-9	20A9	20A-7	20A7	20A-3	20A3	20A-1	20A1
	Type
2420.b.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
2420.b.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
2420.b.1c1	C	$1$	$-1$	$-i$	$i$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$i$	$-i$	$-i$	$-i$	$i$	$-i$	$i$	$i$
2420.b.1c2	C	$1$	$-1$	$i$	$-i$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-i$	$i$	$i$	$i$	$-i$	$i$	$-i$	$-i$
2420.b.1d1	C	$1$	$1$	$1$	$1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^2$	${\zeta }_5^{-2}$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_5^{-2}$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^2$
2420.b.1d2	C	$1$	$1$	$1$	$1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^{-2}$	${\zeta }_5^2$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_5^2$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5^{-2}$
2420.b.1d3	C	$1$	$1$	$1$	$1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5$	${\zeta }_5^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_5^{-1}$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5$
2420.b.1d4	C	$1$	$1$	$1$	$1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5^{-1}$	${\zeta }_5$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_5$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-1}$
2420.b.1e1	C	$1$	$1$	$-1$	$-1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^2$	${\zeta }_5^{-2}$	$1$	$1$	$1$	$1$	$1$	$1$	$-{\zeta }_5^{-2}$	$-{\zeta }_5$	$-{\zeta }_5^{-1}$	$-{\zeta }_5^{-2}$	$-{\zeta }_5^{-1}$	$-{\zeta }_5^2$	$-{\zeta }_5$	$-{\zeta }_5^2$
2420.b.1e2	C	$1$	$1$	$-1$	$-1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^{-2}$	${\zeta }_5^2$	$1$	$1$	$1$	$1$	$1$	$1$	$-{\zeta }_5^2$	$-{\zeta }_5^{-1}$	$-{\zeta }_5$	$-{\zeta }_5^2$	$-{\zeta }_5$	$-{\zeta }_5^{-2}$	$-{\zeta }_5^{-1}$	$-{\zeta }_5^{-2}$
2420.b.1e3	C	$1$	$1$	$-1$	$-1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5$	${\zeta }_5^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$	$-{\zeta }_5^{-1}$	$-{\zeta }_5^{-2}$	$-{\zeta }_5^2$	$-{\zeta }_5^{-1}$	$-{\zeta }_5^2$	$-{\zeta }_5$	$-{\zeta }_5^{-2}$	$-{\zeta }_5$
2420.b.1e4	C	$1$	$1$	$-1$	$-1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5^{-1}$	${\zeta }_5$	$1$	$1$	$1$	$1$	$1$	$1$	$-{\zeta }_5$	$-{\zeta }_5^2$	$-{\zeta }_5^{-2}$	$-{\zeta }_5$	$-{\zeta }_5^{-2}$	$-{\zeta }_5^{-1}$	$-{\zeta }_5^2$	$-{\zeta }_5^{-1}$
2420.b.1f1	C	$1$	$-1$	$-{\zeta }_{20}^5$	${\zeta }_{20}^5$	$-{\zeta }_{20}^2$	${\zeta }_{20}^8$	${\zeta }_{20}^4$	$-{\zeta }_{20}^6$	${\zeta }_{20}^6$	$-{\zeta }_{20}^4$	$-{\zeta }_{20}^8$	${\zeta }_{20}^2$	$1$	$1$	$1$	$1$	$1$	$1$	$-{\zeta }_{20}^7$	$-{\zeta }_{20}^9$	$-{\zeta }_{20}$	${\zeta }_{20}^7$	${\zeta }_{20}$	${\zeta }_{20}^3$	${\zeta }_{20}^9$	$-{\zeta }_{20}^3$
2420.b.1f2	C	$1$	$-1$	${\zeta }_{20}^5$	$-{\zeta }_{20}^5$	${\zeta }_{20}^8$	$-{\zeta }_{20}^2$	$-{\zeta }_{20}^6$	${\zeta }_{20}^4$	$-{\zeta }_{20}^4$	${\zeta }_{20}^6$	${\zeta }_{20}^2$	$-{\zeta }_{20}^8$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_{20}^3$	${\zeta }_{20}$	${\zeta }_{20}^9$	$-{\zeta }_{20}^3$	$-{\zeta }_{20}^9$	$-{\zeta }_{20}^7$	$-{\zeta }_{20}$	${\zeta }_{20}^7$
2420.b.1f3	C	$1$	$-1$	$-{\zeta }_{20}^5$	${\zeta }_{20}^5$	${\zeta }_{20}^8$	$-{\zeta }_{20}^2$	$-{\zeta }_{20}^6$	${\zeta }_{20}^4$	$-{\zeta }_{20}^4$	${\zeta }_{20}^6$	${\zeta }_{20}^2$	$-{\zeta }_{20}^8$	$1$	$1$	$1$	$1$	$1$	$1$	$-{\zeta }_{20}^3$	$-{\zeta }_{20}$	$-{\zeta }_{20}^9$	${\zeta }_{20}^3$	${\zeta }_{20}^9$	${\zeta }_{20}^7$	${\zeta }_{20}$	$-{\zeta }_{20}^7$
2420.b.1f4	C	$1$	$-1$	${\zeta }_{20}^5$	$-{\zeta }_{20}^5$	$-{\zeta }_{20}^2$	${\zeta }_{20}^8$	${\zeta }_{20}^4$	$-{\zeta }_{20}^6$	${\zeta }_{20}^6$	$-{\zeta }_{20}^4$	$-{\zeta }_{20}^8$	${\zeta }_{20}^2$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_{20}^7$	${\zeta }_{20}^9$	${\zeta }_{20}$	$-{\zeta }_{20}^7$	$-{\zeta }_{20}$	$-{\zeta }_{20}^3$	$-{\zeta }_{20}^9$	${\zeta }_{20}^3$
2420.b.1f5	C	$1$	$-1$	$-{\zeta }_{20}^5$	${\zeta }_{20}^5$	$-{\zeta }_{20}^6$	${\zeta }_{20}^4$	$-{\zeta }_{20}^2$	${\zeta }_{20}^8$	$-{\zeta }_{20}^8$	${\zeta }_{20}^2$	$-{\zeta }_{20}^4$	${\zeta }_{20}^6$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_{20}$	${\zeta }_{20}^7$	${\zeta }_{20}^3$	$-{\zeta }_{20}$	$-{\zeta }_{20}^3$	$-{\zeta }_{20}^9$	$-{\zeta }_{20}^7$	${\zeta }_{20}^9$
2420.b.1f6	C	$1$	$-1$	${\zeta }_{20}^5$	$-{\zeta }_{20}^5$	${\zeta }_{20}^4$	$-{\zeta }_{20}^6$	${\zeta }_{20}^8$	$-{\zeta }_{20}^2$	${\zeta }_{20}^2$	$-{\zeta }_{20}^8$	${\zeta }_{20}^6$	$-{\zeta }_{20}^4$	$1$	$1$	$1$	$1$	$1$	$1$	$-{\zeta }_{20}^9$	$-{\zeta }_{20}^3$	$-{\zeta }_{20}^7$	${\zeta }_{20}^9$	${\zeta }_{20}^7$	${\zeta }_{20}$	${\zeta }_{20}^3$	$-{\zeta }_{20}$
2420.b.1f7	C	$1$	$-1$	$-{\zeta }_{20}^5$	${\zeta }_{20}^5$	${\zeta }_{20}^4$	$-{\zeta }_{20}^6$	${\zeta }_{20}^8$	$-{\zeta }_{20}^2$	${\zeta }_{20}^2$	$-{\zeta }_{20}^8$	${\zeta }_{20}^6$	$-{\zeta }_{20}^4$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_{20}^9$	${\zeta }_{20}^3$	${\zeta }_{20}^7$	$-{\zeta }_{20}^9$	$-{\zeta }_{20}^7$	$-{\zeta }_{20}$	$-{\zeta }_{20}^3$	${\zeta }_{20}$
2420.b.1f8	C	$1$	$-1$	${\zeta }_{20}^5$	$-{\zeta }_{20}^5$	$-{\zeta }_{20}^6$	${\zeta }_{20}^4$	$-{\zeta }_{20}^2$	${\zeta }_{20}^8$	$-{\zeta }_{20}^8$	${\zeta }_{20}^2$	$-{\zeta }_{20}^4$	${\zeta }_{20}^6$	$1$	$1$	$1$	$1$	$1$	$1$	$-{\zeta }_{20}$	$-{\zeta }_{20}^7$	$-{\zeta }_{20}^3$	${\zeta }_{20}$	${\zeta }_{20}^3$	${\zeta }_{20}^9$	${\zeta }_{20}^7$	$-{\zeta }_{20}^9$
2420.b.20a	R	$20$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$-2$	$-2$	$-2$	$-2$	$9$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
2420.b.20b	R	$20$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$-2$	$-2$	$-2$	$9$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
2420.b.20c	R	$20$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$-2$	$-2$	$9$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
2420.b.20d	R	$20$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$-2$	$9$	$-2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
2420.b.20e	R	$20$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$9$	$-2$	$-2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
2420.b.20f	R	$20$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$9$	$-2$	$-2$	$-2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed