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Properties

Label	45T49
Degree	$45$
Order	$360$
Cyclic	no
Abelian	no
Solvable	no
Primitive	no
$p$-group	no
Group:	$A_6$

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Show commands: Magma

magma: G := TransitiveGroup(45, 49);

Group action invariants

Degree $n$:		$45$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$49$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$A_6$
Parity:		$1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$1$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(1,2)(4,44,15,27)(5,43,13,25)(6,45,14,26)(7,11,38,29)(8,10,39,30)(9,12,37,28)(16,24)(17,22,18,23)(19,35,40,32)(20,36,41,31)(21,34,42,33), (1,25,34,45,32)(2,26,36,44,33)(3,27,35,43,31)(4,16,23,42,38)(5,18,22,40,37)(6,17,24,41,39)(7,10,13,21,28)(8,12,14,19,29)(9,11,15,20,30)	magma: Generators(G);

Low degree resolvents

none
Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None
Degree 5: None
Degree 9: None
Degree 15: $A_6$ x 2

Low degree siblings

6T15 x 2, 10T26, 15T20 x 2, 20T89, 30T88 x 2, 36T555, 40T304
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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $	$45$	$2$	$( 4,15)( 5,13)( 6,14)( 7,38)( 8,39)( 9,37)(10,30)(11,29)(12,28)(17,18)(19,40) (20,41)(21,42)(22,23)(25,43)(26,45)(27,44)(31,36)(32,35)(33,34)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 1 $	$90$	$4$	$( 2, 3)( 4,32,20,43)( 5,33,19,45)( 6,31,21,44)( 7,23,39,18)( 8,22,38,17) ( 9,24,37,16)(10,29,11,30)(12,28)(13,26,40,34)(14,27,42,36)(15,25,41,35)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$40$	$3$	$( 1, 2, 3)( 4, 5, 6)( 7,22,30)( 8,23,29)( 9,24,28)(10,39,18)(11,38,17) (12,37,16)(13,21,41)(14,20,40)(15,19,42)(25,44,33)(26,43,31)(27,45,32) (34,35,36)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $	$72$	$5$	$( 1, 4, 9,11,32)( 2, 5, 8,12,31)( 3, 6, 7,10,33)(13,36,42,23,44) (14,34,40,24,43)(15,35,41,22,45)(16,38,25,19,30)(17,37,26,21,28) (18,39,27,20,29)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$40$	$3$	$( 1, 4,35)( 2, 5,36)( 3, 6,34)( 7,25,14)( 8,27,13)( 9,26,15)(10,16,38) (11,18,37)(12,17,39)(19,24,43)(20,22,44)(21,23,45)(28,31,42)(29,32,41) (30,33,40)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $	$72$	$5$	$( 1, 4,14,42,20)( 2, 6,13,40,21)( 3, 5,15,41,19)( 7,39,44,28,31) ( 8,37,43,29,33)( 9,38,45,30,32)(10,16,26,36,23)(11,18,27,34,24) (12,17,25,35,22)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$360=2^{3} \cdot 3^{2} \cdot 5$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		no	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		360.118	magma: IdentifyGroup(G);
Character table:

		1A	2A	3A	3B	4A	5A1	5A2
	Size	1	45	40	40	90	72	72
	2 P	1A	1A	3A	3B	2A	5A2	5A1
	3 P	1A	2A	1A	1A	4A	5A2	5A1
	5 P	1A	2A	3A	3B	4A	1A	1A
	Type
360.118.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
360.118.5a	R	$5$	$1$	$- 1$	$2$	$- 1$	$0$	$0$
360.118.5b	R	$5$	$1$	$2$	$- 1$	$- 1$	$0$	$0$
360.118.8a1	R	$8$	$0$	$- 1$	$- 1$	$0$	$- ζ_{5}^{- 1} - ζ_{5}$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$
360.118.8a2	R	$8$	$0$	$- 1$	$- 1$	$0$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$- ζ_{5}^{- 1} - ζ_{5}$
360.118.9a	R	$9$	$1$	$0$	$0$	$1$	$- 1$	$- 1$
360.118.10a	R	$10$	$- 2$	$1$	$1$	$0$	$0$	$0$

magma: CharacterTable(G);