Galois group: 20T15

• Label: 20T15
• Degree: $20$
• Order: $60$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $A_5$

Group action invariants

Degree $n$:		$20$
Transitive number $t$:		$15$
Group:		$A_5$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,17)(2,18)(3,4)(5,7)(6,8)(9,20)(10,19)(11,14)(12,13)(15,16), (1,6,10,13,17)(2,5,9,14,18)(3,8,12,15,20)(4,7,11,16,19)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $A_5$

Degree 10: $A_{5}$

Low degree siblings

5T4, 6T12, 10T7, 12T33, 15T5, 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	$1^{20}$	$1$	$1$	$0$	$()$
2A	$2^{10}$	$15$	$2$	$10$	$( 1, 9)( 2,10)( 3, 7)( 4, 8)( 5, 6)(11,20)(12,19)(13,18)(14,17)(15,16)$
3A	$3^{6},1^{2}$	$20$	$3$	$12$	$( 1, 3, 7)( 2, 4, 8)( 5,14,19)( 6,13,20)(11,15,17)(12,16,18)$
5A1	$5^{4}$	$12$	$5$	$16$	$( 1,11, 4,13,18)( 2,12, 3,14,17)( 5,19,16, 9, 8)( 6,20,15,10, 7)$
5A2	$5^{4}$	$12$	$5$	$16$	$( 1,17, 8,19,12)( 2,18, 7,20,11)( 3, 5,14,15,10)( 4, 6,13,16, 9)$

Malle's constant $a(G)$:     $1/10$

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