Galois group: 5T5

• Label: 5T5
• Degree: $5$
• Order: $120$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $S_5$

Group invariants

Abstract group:		$S_5$
Order:		$120=2^{3} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$5$
Transitive number $t$:		$5$
CHM label:		$S5$
Parity:		$-1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,2)$, $(1,2,3,4,5)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62

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	$1^{5}$	$1$	$1$	$0$	$()$
2A	$2,1^{3}$	$10$	$2$	$1$	$(2,4)$
2B	$2^{2},1$	$15$	$2$	$2$	$(1,4)(3,5)$
3A	$3,1^{2}$	$20$	$3$	$2$	$(1,3,5)$
4A	$4,1$	$30$	$4$	$3$	$(1,3,4,5)$
5A	$5$	$24$	$5$	$4$	$(1,4,2,5,3)$
6A	$3,2$	$20$	$6$	$3$	$(1,5,3)(2,4)$

Malle's constant $a(G)$:     $1$

Character table

		1A	2A	2B	3A	4A	5A	6A
	Size	1	10	15	20	30	24	20
	2 P	1A	1A	1A	3A	2B	5A	3A
	3 P	1A	2A	2B	1A	4A	5A	2A
	5 P	1A	2A	2B	3A	4A	1A	6A
	Type
120.34.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
120.34.1b	R	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
120.34.4a	R	$4$	$2$	$0$	$1$	$0$	$-1$	$-1$
120.34.4b	R	$4$	$-2$	$0$	$1$	$0$	$-1$	$1$
120.34.5a	R	$5$	$-1$	$1$	$-1$	$1$	$0$	$-1$
120.34.5b	R	$5$	$1$	$1$	$-1$	$-1$	$0$	$1$
120.34.6a	R	$6$	$0$	$-2$	$0$	$0$	$1$	$0$

Regular extensions

$f_{ 1 } =$	$x^{5}+s x^{3}+t x+t$
	The polynomial $f_{1}$ is generic for the base field $\Q$