Properties

 Label 8T5 Order $$8$$ n $$8$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $Q_8$

Related objects

Group action invariants

 Degree $n$ : $8$ Transitive number $t$ : $5$ Group : $Q_8$ CHM label : $Q_{8}(8)$ Parity: $1$ Primitive: No Generators: (1,2,3,8)(4,5,6,7), (1,7,3,5)(2,6,8,4) $|\Aut(F/K)|$: $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $V_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $V_4$

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 $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4$ $2$ $4$ $(1,2,3,8)(4,5,6,7)$ $2, 2, 2, 2$ $1$ $2$ $(1,3)(2,8)(4,6)(5,7)$ $4, 4$ $2$ $4$ $(1,4,3,6)(2,7,8,5)$ $4, 4$ $2$ $4$ $(1,5,3,7)(2,4,8,6)$

Group invariants

 Order: $8=2^{3}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [8, 4]
 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 . .