Galois group: 22T4

• Label: 22T4
• Degree: $22$
• Order: $110$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $F_{11}$

Group invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $F_{11}$

Low degree siblings

11T4

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

Malle's constant $a(G)$:     $1/11$

Character table

1A

2A

5A1

5A-1

5A2

5A-2

10A1

10A-1

10A3

10A-3

11A

Size

1

11

11

11

11

11

11

11

11

11

10

2 P

1A

1A

5A2

5A-2

5A-1

5A1

5A1

5A-1

5A-2

5A2

11A

5 P

1A

2A

1A

1A

1A

1A

2A

2A

2A

2A

11A

11 P

1A

2A

5A1

5A-1

5A2

5A-2

10A1

10A-1

10A3

10A-3

1A

Type

110.1.1a

R

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

110.1.1b

R

$1$

$-1$

$1$

$1$

$1$

$1$

$-1$

$-1$

$-1$

$-1$

$1$

110.1.1c1

C

$1$

$1$

${\zeta }_5^{-2}$

${\zeta }_5^2$

${\zeta }_5^{-1}$

${\zeta }_5$

${\zeta }_5^{-2}$

${\zeta }_5$

${\zeta }_5^2$

${\zeta }_5^{-1}$

$1$

110.1.1c2

C

$1$

$1$

${\zeta }_5^2$

${\zeta }_5^{-2}$

${\zeta }_5$

${\zeta }_5^{-1}$

${\zeta }_5^2$

${\zeta }_5^{-1}$

${\zeta }_5^{-2}$

${\zeta }_5$

$1$

110.1.1c3

C

$1$

$1$

${\zeta }_5^{-1}$

${\zeta }_5$

${\zeta }_5^2$

${\zeta }_5^{-2}$

${\zeta }_5^{-1}$

${\zeta }_5^{-2}$

${\zeta }_5$

${\zeta }_5^2$

$1$

110.1.1c4

C

$1$

$1$

${\zeta }_5$

${\zeta }_5^{-1}$

${\zeta }_5^{-2}$

${\zeta }_5^2$

${\zeta }_5$

${\zeta }_5^2$

${\zeta }_5^{-1}$

${\zeta }_5^{-2}$

$1$

110.1.1d1

C

$1$

$-1$

${\zeta }_5^{-2}$

${\zeta }_5^2$

${\zeta }_5^{-1}$

${\zeta }_5$

$-{\zeta }_5^{-2}$

$-{\zeta }_5$

$-{\zeta }_5^2$

$-{\zeta }_5^{-1}$

$1$

110.1.1d2

C

$1$

$-1$

${\zeta }_5^2$

${\zeta }_5^{-2}$

${\zeta }_5$

${\zeta }_5^{-1}$

$-{\zeta }_5^2$

$-{\zeta }_5^{-1}$

$-{\zeta }_5^{-2}$

$-{\zeta }_5$

$1$

110.1.1d3

C

$1$

$-1$

${\zeta }_5^{-1}$

${\zeta }_5$

${\zeta }_5^2$

${\zeta }_5^{-2}$

$-{\zeta }_5^{-1}$

$-{\zeta }_5^{-2}$

$-{\zeta }_5$

$-{\zeta }_5^2$

$1$

110.1.1d4

C

$1$

$-1$

${\zeta }_5$

${\zeta }_5^{-1}$

${\zeta }_5^{-2}$

${\zeta }_5^2$

$-{\zeta }_5$

$-{\zeta }_5^2$

$-{\zeta }_5^{-1}$

$-{\zeta }_5^{-2}$

$1$

110.1.10a

R

$10$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$-1$

Regular extensions

Data not computed