Galois group: 4T5

• Label: 4T5
• Degree: $4$
• Order: $24$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $S_4$

Group invariants

Abstract group:		$S_4$
Order:		$24=2^{3} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$4$
Transitive number $t$:		$5$
CHM label:		$S4$
Parity:		$-1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,2,3,4)$, $(1,2)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$6$:	$S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Low degree siblings

6T7, 6T8, 8T14, 12T8, 12T9, 24T10

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	$1^{4}$	$1$	$1$	$0$	$()$
2A	$2^{2}$	$3$	$2$	$2$	$(1,2)(3,4)$
2B	$2,1^{2}$	$6$	$2$	$1$	$(1,2)$
3A	$3,1$	$8$	$3$	$2$	$(1,2,3)$
4A	$4$	$6$	$4$	$3$	$(1,4,2,3)$

Malle's constant $a(G)$:     $1$

Character table

		1A	2A	2B	3A	4A
	Size	1	3	6	8	6
	2 P	1A	1A	1A	3A	2A
	3 P	1A	2A	2B	1A	4A
	Type
24.12.1a	R	$1$	$1$	$1$	$1$	$1$
24.12.1b	R	$1$	$1$	$-1$	$1$	$-1$
24.12.2a	R	$2$	$2$	$0$	$-1$	$0$
24.12.3a	R	$3$	$-1$	$1$	$0$	$-1$
24.12.3b	R	$3$	$-1$	$-1$	$0$	$1$

Regular extensions

$f_{ 1 } =$	$x^{4}+s x^{2}+t x+t$
	The polynomial $f_{1}$ is generic for any base field $K$