Group action invariants
Degree $n$: | $8$ | |
Transitive number $t$: | $1$ | |
Group: | $C_8$ | |
CHM label: | $C(8)=8$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $1$ | |
$|\Aut(F/K)|$: | $8$ | |
Generators: | (1,2,3,4,5,6,7,8) |
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) |