# Properties

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

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(8, 50);

## Group action invariants

 Degree $n$: $8$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $50$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_8$ CHM label: $S8$ Parity: $-1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2,3,4,5,6,7,8), (1,2) magma: Generators(G);

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $40320=2^{7} \cdot 3^{2} \cdot 5 \cdot 7$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Label: 40320.a magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);