Galois group: 5T4

• Label: 5T4
• Degree: $5$
• Order: $60$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $A_5$

This is the smallest instance of a non-solvable Galois group, implying that the general quintic cannot be solvable in radicals and motivating the creation of Galois theory.

Group action invariants

Degree $n$:		$5$
Transitive number $t$:		$4$
Group:		$A_5$
CHM label:		$A5$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,3), (3,4,5)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

6T12, 10T7, 12T33, 15T5, 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 $	$1$	$1$	$()$
	$ 3, 1, 1 $	$20$	$3$	$(3,4,5)$
	$ 2, 2, 1 $	$15$	$2$	$(2,3)(4,5)$
	$ 5 $	$12$	$5$	$(1,2,3,4,5)$
	$ 5 $	$12$	$5$	$(1,2,3,5,4)$

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