Galois group: 40T62

• Label: 40T62
• Degree: $40$
• Order: $120$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $S_5$

Group invariants

Abstract group:		$S_5$
Order:		$120=2^{3} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$40$
Transitive number $t$:		$62$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(1,29,19,2,30,20)(3,32,17,4,31,18)(5,25,35,13,37,23)(6,26,36,14,38,24)(7,27,33,16,39,22)(8,28,34,15,40,21)(9,12)(10,11)$, $(1,18,14)(2,17,13)(3,20,15)(4,19,16)(9,39,33)(10,40,34)(11,38,36)(12,37,35)(21,25,31)(22,26,32)(23,28,29)(24,27,30)$

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 4: None

Degree 5: $S_5$

Degree 8: None

Degree 10: $S_5$, $S_5$

Degree 20: 20T30, 20T32, 20T35

Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27

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	$1^{40}$	$1$	$1$	$0$	$()$
2A	$2^{20}$	$10$	$2$	$20$	$( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,16)( 8,15)( 9,12)(10,11)(17,18)(19,20)(21,34)(22,33)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,30)(31,32)$
2B	$2^{20}$	$15$	$2$	$20$	$( 1,35)( 2,36)( 3,33)( 4,34)( 5,32)( 6,31)( 7,29)( 8,30)( 9,25)(10,26)(11,28)(12,27)(13,23)(14,24)(15,21)(16,22)(17,20)(18,19)(37,40)(38,39)$
3A	$3^{12},1^{4}$	$20$	$3$	$24$	$( 1,19,30)( 2,20,29)( 3,17,31)( 4,18,32)( 5,35,37)( 6,36,38)( 7,33,39)( 8,34,40)(13,23,25)(14,24,26)(15,21,28)(16,22,27)$
4A	$4^{10}$	$30$	$4$	$30$	$( 1, 7,35,29)( 2, 8,36,30)( 3, 5,33,32)( 4, 6,34,31)( 9,24,25,14)(10,23,26,13)(11,22,28,16)(12,21,27,15)(17,40,20,37)(18,39,19,38)$
5A	$5^{8}$	$24$	$5$	$32$	$( 1,37,10,22,19)( 2,38, 9,21,20)( 3,39,11,23,17)( 4,40,12,24,18)( 5,26,34,16,32)( 6,25,33,15,31)( 7,28,36,13,29)( 8,27,35,14,30)$
6A	$6^{6},2^{2}$	$20$	$6$	$32$	$( 1,29,19, 2,30,20)( 3,32,17, 4,31,18)( 5,25,35,13,37,23)( 6,26,36,14,38,24)( 7,27,33,16,39,22)( 8,28,34,15,40,21)( 9,12)(10,11)$

Malle's constant $a(G)$:     $1/20$

Character table

		1A	2A	2B	3A	4A	5A	6A
	Size	1	10	15	20	30	24	20
	2 P	1A	1A	1A	3A	2B	5A	3A
	3 P	1A	2A	2B	1A	4A	5A	2A
	5 P	1A	2A	2B	3A	4A	1A	6A
	Type
120.34.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
120.34.1b	R	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
120.34.4a	R	$4$	$2$	$0$	$1$	$0$	$-1$	$-1$
120.34.4b	R	$4$	$-2$	$0$	$1$	$0$	$-1$	$1$
120.34.5a	R	$5$	$-1$	$1$	$-1$	$1$	$0$	$-1$
120.34.5b	R	$5$	$1$	$1$	$-1$	$-1$	$0$	$1$
120.34.6a	R	$6$	$0$	$-2$	$0$	$0$	$1$	$0$

Regular extensions

Data not computed