Group action invariants
Degree $n$: | $16$ | |
Transitive number $t$: | $6$ | |
Group: | $C_8: C_2$ | |
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), (1,3,5,8,10,12,14,15)(2,4,6,7,9,11,13,16) |
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: $C_2$ x 3
Degree 8: $C_4\times C_2$, $C_8:C_2$
Low degree siblings
8T7Siblings 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 $ | $2$ | $2$ | $( 1, 2)( 3,11)( 4,12)( 5, 6)( 7,15)( 8,16)( 9,10)(13,14)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1, 3, 5, 8,10,12,14,15)( 2, 4, 6, 7, 9,11,13,16)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1, 4,14,16,10,11, 5, 7)( 2, 3,13,15, 9,12, 6, 8)$ |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 5,10,14)( 2, 6, 9,13)( 3, 8,12,15)( 4, 7,11,16)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 6,10,13)( 2, 5, 9,14)( 3,16,12, 7)( 4,15,11, 8)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1, 7, 5,11,10,16,14, 4)( 2, 8, 6,12, 9,15,13, 3)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1, 8,14, 3,10,15, 5,12)( 2, 7,13, 4, 9,16, 6,11)$ |
$ 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 $ | $1$ | $4$ | $( 1,14,10, 5)( 2,13, 9, 6)( 3,15,12, 8)( 4,16,11, 7)$ |
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 |