Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $969$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $4$ | |
| Generators: | (1,3,2,4)(5,7,6,8)(9,15,13,11)(10,16,14,12), (1,13,8,16,6,10,3,11)(2,14,7,15,5,9,4,12), (1,12)(2,11)(3,9,8,14)(4,10,7,13)(5,16)(6,15) | |
| $|\Aut(F/K)|$: | $4$ |
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 12, $C_2^3$ 16: $D_4\times C_2$ x 6, $Q_8:C_2$ 32: $C_2^2 \wr C_2$ x 3, 16T34 x 3, $C_4^2:C_2$ 64: $(((C_4 \times C_2): C_2):C_2):C_2$ x 6, 32T320 128: $C_2 \wr C_2\wr C_2$ x 2, 16T342 x 2, 16T350 x 3 256: 32T5807 x 2, 32T6030 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$, $C_2 \wr C_2\wr C_2$ x 2
Low degree siblings
16T969 x 63, 32T10844 x 32, 32T10845 x 32, 32T10846 x 16, 32T10847 x 16, 32T10848 x 16, 32T19406 x 8, 32T19412 x 8, 32T19429 x 16, 32T20944 x 16, 32T21627 x 8, 32T21633 x 16, 32T21675 x 16, 32T21677 x 8Siblings 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 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,13)(10,14)(11,15)(12,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,14)(10,13)(11,16)(12,15)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,15,13,11)(10,16,14,12)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 8)( 4, 7)( 9,13)(10,14)(11,12)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 9,13)(10,14)(11,15)(12,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 3, 8)( 4, 7)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,13)(10,14)(11,12)(15,16)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,15,10,16)(11,14,12,13)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 9,14)(10,13)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 7)( 4, 8)(11,12)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,15,14,12)(10,16,13,11)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 7)( 4, 8)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 1, 2)( 3, 8)( 4, 7)( 5, 6)(11,12)(15,16)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,11,14,16)(10,12,13,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,15)(10,16)(11,13)(12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,16)(10,15)(11,14)(12,13)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1,13, 8,16, 6,10, 3,11)( 2,14, 7,15, 5, 9, 4,12)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,11, 3, 9)( 2,12, 4,10)( 5,15, 7,13)( 6,16, 8,14)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1, 9, 4,16, 6,14, 7,11)( 2,10, 3,15, 5,13, 8,12)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,15, 7, 9)( 2,16, 8,10)( 3,13, 5,11)( 4,14, 6,12)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 9,12)(10,11)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,16)(10,15)(11,14)(12,13)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,11)(10,12)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,15)(10,16)(11,13)(12,14)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,10)(11,16)(12,15)(13,14)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 1, 7, 2, 8)( 3, 6, 4, 5)(11,15)(12,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 3, 8)( 4, 7)( 9,11,14,16)(10,12,13,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,15,14,12)(10,16,13,11)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)(11,12)(15,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $32$ | $4$ | $( 1,13, 6,10)( 2,14, 5, 9)( 3,11)( 4,12)( 7,15)( 8,16)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,11, 2,12)( 3, 9, 4,10)( 5,15, 6,16)( 7,13, 8,14)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,15, 5,11)( 2,16, 6,12)( 3,13, 7, 9)( 4,14, 8,10)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 9, 2,10)( 3,12, 7,16)( 4,11, 8,15)( 5,13, 6,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1,15)( 2,16)( 3,10)( 4, 9)( 5,11)( 6,12)( 7,14)( 8,13)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,11, 6,16)( 2,12, 5,15)( 3,14, 8, 9)( 4,13, 7,10)$ |
Group invariants
| Order: | $512=2^{9}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [512, 51562] |
| Character table: Data not available. |