Galois Group: 8T50

• Label: 8T50
• Order: \(40320\)
• n: \(8\)
• 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
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,2,3,4,5,6,7,8), (1,2)
$|\Aut(F/K)|$:		$1$

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

Cycle Type	Size	Order	Representative
$ 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
$ 2, 2, 1, 1, 1, 1 $	$210$	$2$	$(3,7)(4,5)$
$ 3, 1, 1, 1, 1, 1 $	$112$	$3$	$(1,2,8)$
$ 3, 2, 2, 1 $	$1680$	$6$	$(1,8,2)(3,7)(4,5)$
$ 2, 2, 2, 2 $	$105$	$2$	$(1,2)(3,7)(4,5)(6,8)$
$ 4, 4 $	$1260$	$4$	$(1,8,2,6)(3,5,7,4)$
$ 8 $	$5040$	$8$	$( 1, 4, 6, 7, 2, 5, 8, 3)$
$ 2, 2, 2, 1, 1 $	$420$	$2$	$(1,6)(2,5)(3,4)$
$ 3, 3, 1, 1 $	$1120$	$3$	$(1,5,3)(2,4,6)$
$ 6, 1, 1 $	$3360$	$6$	$(1,4,5,6,3,2)$
$ 4, 1, 1, 1, 1 $	$420$	$4$	$( 1, 5, 6, 2)$
$ 4, 3, 1 $	$3360$	$12$	$(1,8,2)(3,4,7,5)$
$ 5, 1, 1, 1 $	$1344$	$5$	$(1,3,4,6,5)$
$ 5, 3 $	$2688$	$15$	$(1,4,5,3,6)(2,8,7)$
$ 2, 1, 1, 1, 1, 1, 1 $	$28$	$2$	$(2,8)$
$ 5, 2, 1 $	$4032$	$10$	$(1,3,4,6,5)(2,8)$
$ 7, 1 $	$5760$	$7$	$(1,8,2,5,7,4,6)$
$ 4, 2, 1, 1 $	$2520$	$4$	$(1,8)(2,3,6,5)$
$ 3, 2, 1, 1, 1 $	$1120$	$6$	$(1,8,6)(4,5)$
$ 4, 2, 2 $	$1260$	$4$	$(1,4,7,2)(3,5)(6,8)$
$ 6, 2 $	$3360$	$6$	$( 1, 8, 5, 2, 6, 4)( 3, 7)$
$ 3, 3, 2 $	$1120$	$6$	$( 1, 6, 5)( 2, 8, 4)( 3, 7)$

Group invariants

Order:		$40320=2^{7} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:		No
Abelian:		No
Solvable:		No
GAP id:		Data not available

Character table: Data not available.