Group action invariants
| Degree $n$: | $16$ | |
| Transitive number $t$: | $14$ | |
| Group: | $Q_{16}$ | |
| Parity: | $1$ | |
| Primitive: | no | |
| Nilpotency class: | $3$ | |
| $|\Aut(F/K)|$: | $16$ | |
| Generators: | (1,13,2,14)(3,12,4,11)(5,9,6,10)(7,15,8,16), (1,12,2,11)(3,9,4,10)(5,8,6,7)(13,15,14,16) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 8: $D_4$
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, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 8, 8 $ | $2$ | $8$ | $( 1, 3, 5,15, 2, 4, 6,16)( 7,10,11,14, 8, 9,12,13)$ |
| $ 8, 8 $ | $2$ | $8$ | $( 1, 4, 5,16, 2, 3, 6,15)( 7, 9,11,13, 8,10,12,14)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 5, 2, 6)( 3,15, 4,16)( 7,11, 8,12)( 9,13,10,14)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 7, 2, 8)( 3,13, 4,14)( 5,12, 6,11)( 9,16,10,15)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 9, 2,10)( 3, 8, 4, 7)( 5,14, 6,13)(11,16,12,15)$ |
Group invariants
| Order: | $16=2^{4}$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [16, 9] |
| Character table: |
2 4 4 3 3 3 2 2
1a 2a 8a 8b 4a 4b 4c
2P 1a 1a 4a 4a 2a 2a 2a
3P 1a 2a 8b 8a 4a 4b 4c
5P 1a 2a 8b 8a 4a 4b 4c
7P 1a 2a 8a 8b 4a 4b 4c
X.1 1 1 1 1 1 1 1
X.2 1 1 -1 -1 1 -1 1
X.3 1 1 -1 -1 1 1 -1
X.4 1 1 1 1 1 -1 -1
X.5 2 2 . . -2 . .
X.6 2 -2 A -A . . .
X.7 2 -2 -A A . . .
A = -E(8)+E(8)^3
= -Sqrt(2) = -r2
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