Galois group: 45T49

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

Group invariants

Abstract group:		$A_6$
Order:		$360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

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

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	Index	Representative
1A	$1^{45}$	$1$	$1$	$0$	$()$
2A	$2^{20},1^{5}$	$45$	$2$	$20$	$( 1,28)( 2,30)( 3,29)( 4, 6)( 7,41)( 8,42)( 9,40)(13,24)(14,23)(15,22)(16,45)(17,43)(18,44)(19,34)(20,36)(21,35)(25,27)(31,38)(32,39)(33,37)$
3A	$3^{15}$	$40$	$3$	$30$	$( 1, 8,38)( 2, 9,37)( 3, 7,39)( 4,29,19)( 5,30,20)( 6,28,21)(10,31,45)(11,33,44)(12,32,43)(13,34,17)(14,35,16)(15,36,18)(22,26,40)(23,27,41)(24,25,42)$
3B	$3^{15}$	$40$	$3$	$30$	$( 1,40,45)( 2,42,43)( 3,41,44)( 4, 7,34)( 5, 8,36)( 6, 9,35)(10,31,14)(11,32,15)(12,33,13)(16,27,20)(17,25,19)(18,26,21)(22,24,23)(28,30,29)(37,39,38)$
4A	$4^{10},2^{2},1$	$90$	$4$	$32$	$( 1,19,28,34)( 2,21,30,35)( 3,20,29,36)( 4,27, 6,25)( 5,26)( 7,31,41,38)( 8,33,42,37)( 9,32,40,39)(10,12)(13,18,24,44)(14,17,23,43)(15,16,22,45)$
5A1	$5^{9}$	$72$	$5$	$36$	$( 1,20, 5,13,41)( 2,21, 4,15,42)( 3,19, 6,14,40)( 7,33,16,34,38)( 8,31,18,35,37)( 9,32,17,36,39)(10,43,25,30,24)(11,45,27,28,22)(12,44,26,29,23)$
5A2	$5^{9}$	$72$	$5$	$36$	$( 1, 5,41,20,13)( 2, 4,42,21,15)( 3, 6,40,19,14)( 7,16,38,33,34)( 8,18,37,31,35)( 9,17,39,32,36)(10,25,24,43,30)(11,27,22,45,28)(12,26,23,44,29)$

Malle's constant $a(G)$:     $1/20$

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	$8$	$0$	$-1$	$-1$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$
360.118.8a2	R	$8$	$0$	$-1$	$-1$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$
360.118.9a	R	$9$	$1$	$0$	$0$	$1$	$-1$	$-1$
360.118.10a	R	$10$	$-2$	$1$	$1$	$0$	$0$	$0$

Regular extensions

Data not computed