Group action invariants
Degree $n$: | $16$ | |
Transitive number $t$: | $12$ | |
Group: | $QD_{16}$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $3$ | |
$|\Aut(F/K)|$: | $16$ | |
Generators: | (1,6)(2,5)(3,11)(4,12)(7,8)(9,13)(10,14)(15,16), (1,3,5,7,9,12,14,15)(2,4,6,8,10,11,13,16) |
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
Low degree siblings
8T8Siblings 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, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 2)( 3, 8)( 4, 7)( 5,13)( 6,14)( 9,10)(11,15)(12,16)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1, 3, 5, 7, 9,12,14,15)( 2, 4, 6, 8,10,11,13,16)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 4, 9,11)( 2, 3,10,12)( 5,16,14, 8)( 6,15,13, 7)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 5, 9,14)( 2, 6,10,13)( 3, 7,12,15)( 4, 8,11,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,15)( 8,16)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1,12, 5,15, 9, 3,14, 7)( 2,11, 6,16,10, 4,13, 8)$ |
Group invariants
Order: | $16=2^{4}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [16, 8] |
Character table: |
2 4 2 3 2 3 4 3 1a 2a 8a 4a 4b 2b 8b 2P 1a 1a 4b 2b 2b 1a 4b 3P 1a 2a 8a 4a 4b 2b 8b 5P 1a 2a 8b 4a 4b 2b 8a 7P 1a 2a 8b 4a 4b 2b 8a 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 . A . . -2 -A X.7 2 . -A . . -2 A A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 |