Group action invariants
Degree $n$: | $16$ | |
Transitive number $t$: | $11$ | |
Group: | $Q_8 : C_2$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $2$ | |
$|\Aut(F/K)|$: | $16$ | |
Generators: | (1,9)(2,10)(3,4)(5,13)(6,14)(7,8)(11,12)(15,16), (1,5,10,14)(2,6,9,13)(3,7,12,16)(4,8,11,15), (1,7,10,16)(2,8,9,15)(3,5,12,14)(4,6,11,13) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $C_2^3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 7
Degree 4: $C_2^2$ x 7
Low degree siblings
8T11 x 3Siblings 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 $ | $2$ | $2$ | $( 1, 2)( 3,11)( 4,12)( 5, 6)( 7,15)( 8,16)( 9,10)(13,14)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 4,10,11)( 2, 3, 9,12)( 5, 8,14,15)( 6, 7,13,16)$ |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 5,10,14)( 2, 6, 9,13)( 3, 7,12,16)( 4, 8,11,15)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 6,10,13)( 2, 5, 9,14)( 3,15,12, 8)( 4,16,11, 7)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 7,10,16)( 2, 8, 9,15)( 3, 5,12,14)( 4, 6,11,13)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 8)( 2, 7)( 3,13)( 4,14)( 5,11)( 6,12)( 9,16)(10,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$ |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,14,10, 5)( 2,13, 9, 6)( 3,16,12, 7)( 4,15,11, 8)$ |
Group invariants
Order: | $16=2^{4}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [16, 13] |
Character table: |
2 4 3 3 3 4 3 3 3 4 4 1a 2a 2b 4a 4b 4c 4d 2c 2d 4e 2P 1a 1a 1a 2d 2d 2d 2d 1a 1a 2d 3P 1a 2a 2b 4a 4e 4c 4d 2c 2d 4b 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 1 -1 1 -1 1 -1 1 1 X.6 1 1 -1 -1 -1 -1 1 1 1 -1 X.7 1 1 -1 -1 1 1 -1 -1 1 1 X.8 1 1 1 1 -1 -1 -1 -1 1 -1 X.9 2 . . . A . . . -2 -A X.10 2 . . . -A . . . -2 A A = -2*E(4) = -2*Sqrt(-1) = -2i |