# Properties

 Label 8T5 Degree $8$ Order $8$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $Q_8$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(8, 5);

## Group action invariants

 Degree $n$: $8$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $5$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $Q_8$ CHM label: $Q_{8}(8)$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $8$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2,3,8)(4,5,6,7), (1,7,3,5)(2,6,8,4) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4$ $2$ $4$ $(1,2,3,8)(4,5,6,7)$ $2, 2, 2, 2$ $1$ $2$ $(1,3)(2,8)(4,6)(5,7)$ $4, 4$ $2$ $4$ $(1,4,3,6)(2,7,8,5)$ $4, 4$ $2$ $4$ $(1,5,3,7)(2,4,8,6)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $8=2^{3}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $2$ Label: 8.4 magma: IdentifyGroup(G);
 Character table:  2 3 2 3 2 2 1a 4a 2a 4b 4c 2P 1a 2a 1a 2a 2a 3P 1a 4a 2a 4b 4c X.1 1 1 1 1 1 X.2 1 -1 1 -1 1 X.3 1 -1 1 1 -1 X.4 1 1 1 -1 -1 X.5 2 . -2 . . 

magma: CharacterTable(G);