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Properties

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

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magma: G := TransitiveGroup(40, 62);

Group action invariants

Degree $n$:		$40$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$62$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$S_5$
Parity:		$1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$4$	magma: Order(Centralizer(SymmetricGroup(n), G));
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)	magma: Generators(G);

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

magma: ConjugacyClasses(G);

Group invariants

Order:		$120=2^{3} \cdot 3 \cdot 5$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		no	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		120.34	magma: IdentifyGroup(G);
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	$6$	$0$	$- 2$	$0$	$0$	$1$	$0$

magma: CharacterTable(G);