Group action invariants
| Degree $n$ : | $8$ | |
| Transitive number $t$ : | $1$ | |
| Group : | $C_8$ | |
| CHM label : | $C(8)=8$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $1$ | |
| Generators: | (1,2,3,4,5,6,7,8) | |
| $|\Aut(F/K)|$: | $8$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 4: $C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
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 $ | $1$ | $1$ | $()$ |
| $ 8 $ | $1$ | $8$ | $(1,2,3,4,5,6,7,8)$ |
| $ 4, 4 $ | $1$ | $4$ | $(1,3,5,7)(2,4,6,8)$ |
| $ 8 $ | $1$ | $8$ | $(1,4,7,2,5,8,3,6)$ |
| $ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ |
| $ 8 $ | $1$ | $8$ | $(1,6,3,8,5,2,7,4)$ |
| $ 4, 4 $ | $1$ | $4$ | $(1,7,5,3)(2,8,6,4)$ |
| $ 8 $ | $1$ | $8$ | $(1,8,7,6,5,4,3,2)$ |
Group invariants
| Order: | $8=2^{3}$ | |
| Cyclic: | Yes | |
| Abelian: | Yes | |
| Solvable: | Yes | |
| GAP id: | [8, 1] |
| Character table: |
2 3 3 3 3 3 3 3 3
1a 8a 4a 8b 2a 8c 4b 8d
X.1 1 1 1 1 1 1 1 1
X.2 1 -1 1 -1 1 -1 1 -1
X.3 1 A -1 -A 1 A -1 -A
X.4 1 -A -1 A 1 -A -1 A
X.5 1 B A -/B -1 -B -A /B
X.6 1 -B A /B -1 B -A -/B
X.7 1 -/B -A B -1 /B A -B
X.8 1 /B -A -B -1 -/B A B
A = E(4)
= Sqrt(-1) = i
B = E(8)
|