Galois group: 22T13

• Label: 22T13
• Degree: $22$
• Order: $1320$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no

Group action invariants

Degree $n$:		$22$
Transitive number $t$:		$13$
Parity:		$-1$
Primitive:		no
Nilpotency class:		$-1$ (not nilpotent)
$|\Aut(F/K)|$:		$2$
Generators:		(1,20,12,5,22,2,19,11,6,21)(3,17,13,8,16,4,18,14,7,15)(9,10), (1,21,15,3,8,10)(2,22,16,4,7,9)(5,14,11,6,13,12)(17,20)(18,19)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$660$:	$\PSL(2,11)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $\PSL(2,11)$

Low degree siblings

22T13, 24T2948

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

Cycle Type	Size	Order	Representative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $	$110$	$3$	$( 3,13,10)( 4,14, 9)( 5,21,11)( 6,22,12)( 7,18,16)( 8,17,15)$
$ 6, 6, 6, 2, 2 $	$110$	$6$	$( 1, 2)( 3,14,10, 4,13, 9)( 5,22,11, 6,21,12)( 7,17,16, 8,18,15)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$55$	$2$	$( 3,10)( 4, 9)( 5,16)( 6,15)( 7,11)( 8,12)(17,22)(18,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$55$	$2$	$( 1, 2)( 3, 9)( 4,10)( 5,15)( 6,16)( 7,12)( 8,11)(13,14)(17,21)(18,22)(19,20)$
$ 10, 10, 2 $	$132$	$10$	$( 1, 2)( 3,17, 5, 8,11, 4,18, 6, 7,12)( 9,16,19,21,14,10,15,20,22,13)$
$ 5, 5, 5, 5, 1, 1 $	$132$	$5$	$( 3,18, 5, 7,11)( 4,17, 6, 8,12)( 9,15,19,22,14)(10,16,20,21,13)$
$ 10, 10, 2 $	$132$	$10$	$( 1, 2)( 3, 9,18,19,21, 4,10,17,20,22)( 5,14,16,12, 7, 6,13,15,11, 8)$
$ 5, 5, 5, 5, 1, 1 $	$132$	$5$	$( 3,10,18,20,21)( 4, 9,17,19,22)( 5,13,16,11, 7)( 6,14,15,12, 8)$
$ 6, 6, 6, 2, 2 $	$110$	$6$	$( 1,20)( 2,19)( 3, 9,13, 4,10,14)( 5, 8,21,17,11,15)( 6, 7,22,18,12,16)$
$ 6, 6, 3, 3, 2, 2 $	$110$	$6$	$( 1,19)( 2,20)( 3,10,13)( 4, 9,14)( 5, 7,21,18,11,16)( 6, 8,22,17,12,15)$
$ 22 $	$60$	$22$	$( 1,20,22,18, 6, 3,15,11, 9,13, 8, 2,19,21,17, 5, 4,16,12,10,14, 7)$
$ 11, 11 $	$60$	$11$	$( 1,19,22,17, 6, 4,15,12, 9,14, 8)( 2,20,21,18, 5, 3,16,11,10,13, 7)$
$ 22 $	$60$	$22$	$( 1,20,22, 3,14,16, 6,10, 8,11,17, 2,19,21, 4,13,15, 5, 9, 7,12,18)$
$ 11, 11 $	$60$	$11$	$( 1,19,22, 4,14,15, 6, 9, 8,12,17)( 2,20,21, 3,13,16, 5,10, 7,11,18)$

Group invariants

Order:		$1320=2^{3} \cdot 3 \cdot 5 \cdot 11$
Cyclic:		no
Abelian:		no
Solvable:		no
GAP id:		[1320, 134]

Character table:

2 3 3 2 2 3 3 1 1 1 1 2 2 1 1 1 1 3 1 1 1 1 1 1 . . . . 1 1 . . . . 5 1 1 . . . . 1 1 1 1 . . . . . . 11 1 1 . . . . . . . . . . 1 1 1 1 1a 2a 3a 6a 2b 2c 10a 5a 10b 5b 6b 6c 22a 11a 22b 11b 2P 1a 1a 3a 3a 1a 1a 5b 5b 5a 5a 3a 3a 11b 11b 11a 11a 3P 1a 2a 1a 2a 2b 2c 10b 5b 10a 5a 2c 2b 22a 11a 22b 11b 5P 1a 2a 3a 6a 2b 2c 2a 1a 2a 1a 6b 6c 22a 11a 22b 11b 7P 1a 2a 3a 6a 2b 2c 10b 5b 10a 5a 6b 6c 22b 11b 22a 11a 11P 1a 2a 3a 6a 2b 2c 10a 5a 10b 5b 6b 6c 2a 1a 2a 1a 13P 1a 2a 3a 6a 2b 2c 10b 5b 10a 5a 6b 6c 22b 11b 22a 11a 17P 1a 2a 3a 6a 2b 2c 10b 5b 10a 5a 6b 6c 22b 11b 22a 11a 19P 1a 2a 3a 6a 2b 2c 10a 5a 10b 5b 6b 6c 22b 11b 22a 11a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 X.3 5 5 -1 -1 1 1 . . . . 1 1 B B /B /B X.4 5 5 -1 -1 1 1 . . . . 1 1 /B /B B B X.5 5 -5 -1 1 1 -1 . . . . -1 1 -B B -/B /B X.6 5 -5 -1 1 1 -1 . . . . -1 1 -/B /B -B B X.7 10 10 1 1 -2 -2 . . . . 1 1 -1 -1 -1 -1 X.8 10 -10 1 -1 -2 2 . . . . -1 1 1 -1 1 -1 X.9 10 10 1 1 2 2 . . . . -1 -1 -1 -1 -1 -1 X.10 10 -10 1 -1 2 -2 . . . . 1 -1 1 -1 1 -1 X.11 11 11 -1 -1 -1 -1 1 1 1 1 -1 -1 . . . . X.12 11 -11 -1 1 -1 1 -1 1 -1 1 1 -1 . . . . X.13 12 -12 . . . . A -A *A -*A . . -1 1 -1 1 X.14 12 -12 . . . . *A -*A A -A . . -1 1 -1 1 X.15 12 12 . . . . -*A -*A -A -A . . 1 1 1 1 X.16 12 12 . . . . -A -A -*A -*A . . 1 1 1 1 A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 B = E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9 = (-1+Sqrt(-11))/2 = b11