Galois group: 15T20

• Label: 15T20
• Degree: $15$
• Order: $360$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $A_6$

Group action invariants

Degree $n$:		$15$
Transitive number $t$:		$20$
Group:		$A_6$
CHM label:		$A_{6}(15)$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8), (1,5)(2,7)(3,6)(4,15)(8,9)(12,13)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: None

Low degree siblings

6T15 x 2, 10T26, 15T20, 20T89, 30T88 x 2, 36T555, 40T304, 45T49

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 $	$1$	$1$	$()$
	$ 3, 3, 3, 3, 1, 1, 1 $	$40$	$3$	$( 3, 8,10)( 4,13, 6)( 5, 7,14)( 9,15,11)$
	$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$45$	$2$	$( 2, 5)( 3,13)( 4,10)( 7,14)( 9,15)(11,12)$
	$ 4, 4, 4, 2, 1 $	$90$	$4$	$( 2, 9, 5,15)( 3,10,13, 4)( 6, 8)( 7,11,14,12)$
	$ 5, 5, 5 $	$72$	$5$	$( 1, 2, 3, 4,11)( 5,12,14, 8,15)( 6,13, 9, 7,10)$
	$ 5, 5, 5 $	$72$	$5$	$( 1, 2, 3, 6, 9)( 4,13,11, 5, 8)( 7,12,14,10,15)$
	$ 3, 3, 3, 3, 3 $	$40$	$3$	$( 1, 2,12)( 3, 9, 5)( 4, 6,13)( 7, 8,15)(10,11,14)$

Group invariants

Order:		$360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		360.118
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	$10$	$-2$	$1$	$1$	$0$	$0$	$0$