# Properties

 Label 8T1 Order $$8$$ n $$8$$ Cyclic Yes Abelian Yes Solvable Yes Primitive No $p$-group Yes Group: $C_8$

# Related objects

## 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)