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	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$15$	$2$	$( 2, 5)( 3,13)( 4,10)( 7,14)( 9,15)(11,12)$
	$ 5, 5, 5 $	$12$	$5$	$( 1, 2, 7,14, 5)( 3,13,15, 8, 9)( 4,11,12,10, 6)$
	$ 5, 5, 5 $	$12$	$5$	$( 1, 2,10,13,11)( 3, 6, 4,15,14)( 5,12, 7, 8, 9)$
	$ 3, 3, 3, 3, 3 $	$20$	$3$	$( 1, 2,12)( 3,11, 7)( 4,13, 6)( 5,10,15)( 8, 9,14)$

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	$5$	$1$	$-1$	$0$	$0$