Group action invariants
Degree $n$ : | $8$ | |
Transitive number $t$ : | $7$ | |
Group : | $C_8:C_2$ | |
CHM label : | $1/2[2^{3}]4$ | |
Parity: | $-1$ | |
Primitive: | No | |
Nilpotency class: | $2$ | |
Generators: | (1,2,3,4,5,6,7,8), (1,5)(3,7) | |
$|\Aut(F/K)|$: | $4$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_4$ x 2, $C_2^2$ 8: $C_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Low degree siblings
16T6Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 1, 1, 1, 1 $ | $2$ | $2$ | $(2,6)(4,8)$ |
$ 8 $ | $2$ | $8$ | $(1,2,3,4,5,6,7,8)$ |
$ 8 $ | $2$ | $8$ | $(1,2,7,8,5,6,3,4)$ |
$ 4, 4 $ | $1$ | $4$ | $(1,3,5,7)(2,4,6,8)$ |
$ 4, 4 $ | $2$ | $4$ | $(1,3,5,7)(2,8,6,4)$ |
$ 8 $ | $2$ | $8$ | $(1,4,3,6,5,8,7,2)$ |
$ 8 $ | $2$ | $8$ | $(1,4,7,2,5,8,3,6)$ |
$ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ |
$ 4, 4 $ | $1$ | $4$ | $(1,7,5,3)(2,8,6,4)$ |
Group invariants
Order: | $16=2^{4}$ | |
Cyclic: | No | |
Abelian: | No | |
Solvable: | Yes | |
GAP id: | [16, 6] |
Character table: |
2 4 3 3 3 4 3 3 3 4 4 1a 2a 8a 8b 4a 4b 8c 8d 2b 4c 2P 1a 1a 4a 4c 2b 2b 4a 4c 1a 2b 3P 1a 2a 8d 8c 4c 4b 8b 8a 2b 4a 5P 1a 2a 8a 8b 4a 4b 8c 8d 2b 4c 7P 1a 2a 8d 8c 4c 4b 8b 8a 2b 4a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 -1 1 -1 1 1 X.3 1 -1 1 -1 1 -1 -1 1 1 1 X.4 1 1 -1 -1 1 1 -1 -1 1 1 X.5 1 -1 A -A -1 1 A -A 1 -1 X.6 1 -1 -A A -1 1 -A A 1 -1 X.7 1 1 A A -1 -1 -A -A 1 -1 X.8 1 1 -A -A -1 -1 A A 1 -1 X.9 2 . . . B . . . -2 -B X.10 2 . . . -B . . . -2 B A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i |