Galois group: 8T50

• Label: 8T50
• Degree: $8$
• Order: $40320$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $S_8$

Group action invariants

Degree $n$:		$8$
Transitive number $t$:		$50$
Group:		$S_8$
CHM label:		$S8$
Parity:		$-1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,3,4,5,6,7,8), (1,2)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Low degree siblings

16T1838, 28T502, 30T1153, 35T44

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
	$1^{8}$	$1$	$1$	$()$
	$2^{2},1^{4}$	$210$	$2$	$(1,7)(3,4)$
	$4,2,1^{2}$	$2520$	$4$	$(1,4,7,3)(5,6)$
	$2^{3},1^{2}$	$420$	$2$	$(1,7)(2,8)(3,4)$
	$2^{4}$	$105$	$2$	$(1,7)(2,8)(3,4)(5,6)$
	$4^{2}$	$1260$	$4$	$(1,4,7,3)(2,6,5,8)$
	$8$	$5040$	$8$	$(1,6,4,5,7,8,3,2)$
	$7,1$	$5760$	$7$	$(1,4,7,3,5,8,2)$
	$3,1^{5}$	$112$	$3$	$(6,8,7)$
	$3,2^{2},1$	$1680$	$6$	$(1,2)(3,5)(6,7,8)$
	$4,1^{4}$	$420$	$4$	$(1,5,2,3)$
	$4,3,1$	$3360$	$12$	$(1,3,2,5)(6,7,8)$
	$2,1^{6}$	$28$	$2$	$(3,8)$
	$5,1^{3}$	$1344$	$5$	$(1,7,4,6,5)$
	$5,2,1$	$4032$	$10$	$(1,6,7,5,4)(3,8)$
	$3^{2},1^{2}$	$1120$	$3$	$(1,3,2)(6,8,7)$
	$6,2$	$3360$	$6$	$(1,7,3,6,2,8)(4,5)$
	$3^{2},2$	$1120$	$6$	$(1,3,2)(4,5)(6,8,7)$
	$6,1^{2}$	$3360$	$6$	$(1,7,2,8,3,6)$
	$5,3$	$2688$	$15$	$(1,3,5,4,2)(6,8,7)$
	$3,2,1^{3}$	$1120$	$6$	$(1,5)(6,7,8)$
	$4,2^{2}$	$1260$	$4$	$(1,2)(3,5)(4,6,8,7)$

Group invariants

Order:		$40320=2^{7} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		40320.a
Character table:

	Size
	2 P
	3 P
	5 P
	7 P
	Type