Galois group: 40T62

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

Group action invariants

Degree $n$:		$40$
Transitive number $t$:		$62$
Group:		$S_5$
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

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

Malle's constant $a(G)$:     $1/20$

Group invariants

Order:		$120=2^{3} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		120.34
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	$6$	$0$	$-2$	$0$	$0$	$1$	$0$