Group action invariants
Degree $n$: | $16$ | |
Transitive number $t$: | $35$ | |
Group: | $C_8:C_2^2$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $3$ | |
$|\Aut(F/K)|$: | $8$ | |
Generators: | (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16), (1,4,6,7,9,12,14,15)(2,3,5,8,10,11,13,16), (1,9)(2,10)(5,13)(6,14) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $D_{4}$ x 2, $C_2^3$ $16$: $D_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 8: $D_4$, $Z_8 : Z_8^\times$ x 2
Low degree siblings
8T15 x 2, 16T38 x 2, 16T45, 32T21Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 3,11)( 4,12)( 7,15)( 8,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 2)( 3, 7)( 4, 8)( 5,14)( 6,13)( 9,10)(11,15)(12,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 2)( 3,15)( 4,16)( 5,14)( 6,13)( 7,11)( 8,12)( 9,10)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,13)( 8,14)( 9,11)(10,12)$ |
$ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 3, 9,11)( 2, 4,10,12)( 5,15,13, 7)( 6,16,14, 8)$ |
$ 8, 8 $ | $4$ | $8$ | $( 1, 4, 6, 7, 9,12,14,15)( 2, 3, 5, 8,10,11,13,16)$ |
$ 8, 8 $ | $4$ | $8$ | $( 1, 4,14,15, 9,12, 6, 7)( 2, 3,13,16,10,11, 5, 8)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 6, 9,14)( 2, 5,10,13)( 3, 8,11,16)( 4, 7,12,15)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 6, 9,14)( 2, 5,10,13)( 3,16,11, 8)( 4,15,12, 7)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)$ |
Group invariants
Order: | $32=2^{5}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [32, 43] |
Character table: |
2 5 4 3 3 3 3 3 3 4 4 5 1a 2a 2b 2c 2d 4a 8a 8b 4b 4c 2e 2P 1a 1a 1a 1a 1a 2e 4b 4b 2e 2e 1a 3P 1a 2a 2b 2c 2d 4a 8a 8b 4b 4c 2e 5P 1a 2a 2b 2c 2d 4a 8a 8b 4b 4c 2e 7P 1a 2a 2b 2c 2d 4a 8a 8b 4b 4c 2e X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 -1 1 1 -1 1 X.3 1 -1 -1 1 1 -1 1 -1 1 -1 1 X.4 1 -1 1 -1 -1 1 1 -1 1 -1 1 X.5 1 -1 1 -1 1 -1 -1 1 1 -1 1 X.6 1 1 -1 -1 -1 -1 1 1 1 1 1 X.7 1 1 -1 -1 1 1 -1 -1 1 1 1 X.8 1 1 1 1 -1 -1 -1 -1 1 1 1 X.9 2 2 . . . . . . -2 -2 2 X.10 2 -2 . . . . . . -2 2 2 X.11 4 . . . . . . . . . -4 |