Galois group: 15T5

• Label: 15T5
• Degree: $15$
• Order: $60$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $A_5$

Group action invariants

Degree $n$:		$15$
Transitive number $t$:		$5$
Group:		$A_5$
CHM label:		$A_{5}(15)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$3$
Generators:		(1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8), (1,4,10)(2,5,8)(3,7,11)(6,9,15)(12,14,13)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $A_5$

Low degree siblings

5T4, 6T12, 10T7, 12T33, 20T15, 30T9

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	$1^{15}$	$1$	$1$	$0$	$()$
2A	$2^{6},1^{3}$	$15$	$2$	$6$	$( 1,12)( 3, 5)( 6,13)( 7,10)( 8,14)(11,15)$
3A	$3^{5}$	$20$	$3$	$10$	$( 1,11, 5)( 2, 9, 4)( 3,10, 6)( 7,15, 8)(12,13,14)$
5A1	$5^{3}$	$12$	$5$	$12$	$( 1,14, 2, 5, 7)( 3, 8,13, 9,15)( 4,10,11, 6,12)$
5A2	$5^{3}$	$12$	$5$	$12$	$( 1,13, 2,11,10)( 3,15, 6,14, 4)( 5, 8,12, 9, 7)$

Malle's constant $a(G)$:     $1/6$

Group invariants

Order:		$60=2^{2} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		60.5
Character table:

		1A	2A	3A	5A1	5A2
	Size	1	15	20	12	12
	2 P	1A	1A	3A	5A2	5A1
	3 P	1A	2A	1A	5A2	5A1
	5 P	1A	2A	3A	1A	1A
	Type
60.5.1a	R	$1$	$1$	$1$	$1$	$1$
60.5.3a1	R	$3$	$-1$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$
60.5.3a2	R	$3$	$-1$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$
60.5.4a	R	$4$	$0$	$1$	$-1$	$-1$
60.5.5a	R	$5$	$1$	$-1$	$0$	$0$