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