Group action invariants
Degree $n$: | $8$ | |
Transitive number $t$: | $4$ | |
Group: | $D_4$ | |
CHM label: | $D_{8}(8)=[4]2$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $2$ | |
$\card{\Aut(F/K)}$: | $8$ | |
Generators: | (1,2,3,8)(4,5,6,7), (1,6)(2,5)(3,4)(7,8) |
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
Low degree siblings
4T3 x 2Siblings are shown 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 $ | $1$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ |
$ 2, 2, 2, 2 $ | $2$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ |
$ 2, 2, 2, 2 $ | $2$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ |
Group invariants
Order: | $8=2^{3}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
Label: | 8.3 |
Character table: |
2 3 2 3 2 2 1a 4a 2a 2b 2c 2P 1a 2a 1a 1a 1a 3P 1a 4a 2a 2b 2c 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 . . |