Galois group: 6T15

• Label: 6T15
• Degree: $6$
• Order: $360$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-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 $n$:		$6$
Transitive number $t$:		$15$
CHM label:		$A6$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,2,3)$, $(1,2)(3,4,5,6)$

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Low degree siblings

6T15, 10T26, 15T20 x 2, 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	Index	Representative
1A	$1^{6}$	$1$	$1$	$0$	$()$
2A	$2^{2},1^{2}$	$45$	$2$	$2$	$(1,4)(3,5)$
3A	$3,1^{3}$	$40$	$3$	$2$	$(2,5,4)$
3B	$3^{2}$	$40$	$3$	$4$	$(1,6,2)(3,4,5)$
4A	$4,2$	$90$	$4$	$4$	$(1,5,4,3)(2,6)$
5A1	$5,1$	$72$	$5$	$4$	$(1,4,6,2,3)$
5A2	$5,1$	$72$	$5$	$4$	$(1,6,3,4,2)$

Malle's constant $a(G)$:     $1/2$

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

$f_{ 1 } =$

$x^{6} + 3 x^{5} + 15 x^{4} + t x^{3} + 12 x - 4$