Galois group: 22T5

• Label: 22T5
• Degree: $22$
• Order: $110$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{11}:C_{10}$

Group invariants

Abstract group:		$C_{11}:C_{10}$
Order:		$110=2 \cdot 5 \cdot 11$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$5$:	$C_5$
$10$:	$C_{10}$
$55$:	$C_{11}:C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $C_{11}:C_5$

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

Malle's constant $a(G)$:     $1/11$

Character table

		1A	2A	5A1	5A-1	5A2	5A-2	10A1	10A-1	10A3	10A-3	11A1	11A-1	22A1	22A-1
	Size	1	1	11	11	11	11	11	11	11	11	5	5	5	5
	2 P	1A	1A	5A2	5A-2	5A-1	5A1	5A1	5A-1	5A-2	5A2	11A-1	11A1	11A1	11A-1
	5 P	1A	2A	1A	1A	1A	1A	2A	2A	2A	2A	11A1	11A-1	22A1	22A-1
	11 P	1A	2A	5A1	5A-1	5A2	5A-2	10A1	10A-1	10A3	10A-3	1A	1A	2A	2A
	Type
110.2.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
110.2.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
110.2.1c1	C	$1$	$1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	$1$	$1$	$1$	$1$
110.2.1c2	C	$1$	$1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5$	$1$	$1$	$1$	$1$
110.2.1c3	C	$1$	$1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5^2$	$1$	$1$	$1$	$1$
110.2.1c4	C	$1$	$1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-2}$	$1$	$1$	$1$	$1$
110.2.1d1	C	$1$	$-1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$	$-{\zeta }_5^2$	$-{\zeta }_5$	$-{\zeta }_5^{-2}$	$-{\zeta }_5^{-1}$	$1$	$1$	$-1$	$-1$
110.2.1d2	C	$1$	$-1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$	$-{\zeta }_5^{-2}$	$-{\zeta }_5^{-1}$	$-{\zeta }_5^2$	$-{\zeta }_5$	$1$	$1$	$-1$	$-1$
110.2.1d3	C	$1$	$-1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$	$-{\zeta }_5$	$-{\zeta }_5^{-2}$	$-{\zeta }_5^{-1}$	$-{\zeta }_5^2$	$1$	$1$	$-1$	$-1$
110.2.1d4	C	$1$	$-1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$	$-{\zeta }_5^{-1}$	$-{\zeta }_5^2$	$-{\zeta }_5$	$-{\zeta }_5^{-2}$	$1$	$1$	$-1$	$-1$
110.2.5a1	C	$5$	$5$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_{11}^{-2}-1-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$	${\zeta }_{11}^{-2}+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$	${\zeta }_{11}^{-2}+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$	$-{\zeta }_{11}^{-2}-1-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$
110.2.5a2	C	$5$	$5$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{11}^{-2}+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$	$-{\zeta }_{11}^{-2}-1-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$	$-{\zeta }_{11}^{-2}-1-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$	${\zeta }_{11}^{-2}+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$
110.2.5b1	C	$5$	$-5$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_{11}^{-2}-1-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$	${\zeta }_{11}^{-2}+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$	$-{\zeta }_{11}^{-2}-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$	${\zeta }_{11}^{-2}+1+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$
110.2.5b2	C	$5$	$-5$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{11}^{-2}+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$	$-{\zeta }_{11}^{-2}-1-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$	${\zeta }_{11}^{-2}+1+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$	$-{\zeta }_{11}^{-2}-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$

Regular extensions

Data not computed