Group action invariants
| Degree $n$ : | $8$ | |
| Transitive number $t$ : | $5$ | |
| Group : | $Q_8$ | |
| CHM label : | $Q_{8}(8)$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $2$ | |
| 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: $C_2^2$ 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 $ | $2$ | $4$ | $(1,5,3,7)(2,4,8,6)$ |
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 . .
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