Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $905$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $5$ | |
| Generators: | (1,6,4,8,2,5,3,7)(9,15,12,13,10,16,11,14), (9,10)(11,12)(13,14)(15,16), (1,12,4,10,2,11,3,9)(5,13,7,16,6,14,8,15), (7,8)(13,15)(14,16) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 15 4: $C_2^2$ x 35 8: $D_{4}$ x 12, $C_2^3$ x 15 16: $D_4\times C_2$ x 18, $C_2^4$ 32: $C_2^2 \wr C_2$ x 4, $C_2^2 \times D_4$ x 3 64: $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T105 128: $C_2 \wr C_2\wr C_2$ x 2, 16T245 256: 16T509 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$
Low degree siblings
16T905 x 7, 32T10487 x 4, 32T10488 x 4, 32T10489 x 4, 32T10490 x 8, 32T10491 x 16, 32T10492 x 8, 32T10493 x 4, 32T10494 x 4, 32T10495 x 4, 32T10496 x 4, 32T20860 x 4, 32T22216 x 8Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 5, 6)( 7, 8)(13,14)(15,16)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 6, 4, 8, 2, 5, 3, 7)( 9,15,12,13,10,16,11,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 4)( 7, 8)( 9,12)(10,11)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 5, 6)( 7, 8)( 9,10)(11,12)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 6, 4, 8, 2, 5, 3, 7)( 9,16,12,14,10,15,11,13)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 4)( 7, 8)( 9,11)(10,12)(13,16)(14,15)$ |
| $ 8, 8 $ | $8$ | $8$ | $( 1,12, 4,10, 2,11, 3, 9)( 5,13, 7,16, 6,14, 8,15)$ |
| $ 8, 8 $ | $8$ | $8$ | $( 1,12, 4,10, 2,11, 3, 9)( 5,14, 7,15, 6,13, 8,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1,13)( 2,14)( 3,15)( 4,16)( 5,10)( 6, 9)( 7,11)( 8,12)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1,16, 2,15)( 3,13, 4,14)( 5,11, 6,12)( 7, 9, 8,10)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,15)( 6,16)( 7,14)( 8,13)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1,15, 3,13, 2,16, 4,14)( 5,12, 7, 9, 6,11, 8,10)$ |
| $ 8, 8 $ | $8$ | $8$ | $( 1,11, 3,10, 2,12, 4, 9)( 5,14, 8,16, 6,13, 7,15)$ |
| $ 8, 8 $ | $8$ | $8$ | $( 1,11, 3,10, 2,12, 4, 9)( 5,13, 8,15, 6,14, 7,16)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1,14, 2,13)( 3,16, 4,15)( 5, 9, 6,10)( 7,12, 8,11)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1,15)( 2,16)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,10, 2, 9)( 3,11, 4,12)( 5,16, 6,15)( 7,13, 8,14)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1,16, 4,13, 2,15, 3,14)( 5,11, 8, 9, 6,12, 7,10)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 5, 8, 6, 7)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,10)(11,12)(13,16,14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,15)(10,16)(11,14)(12,13)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 8, 2, 7)( 3, 6, 4, 5)( 9,13,10,14)(11,15,12,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(15,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 5, 8, 6, 7)( 9,10)(11,12)(13,16,14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,16)(10,15)(11,13)(12,14)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 8, 2, 7)( 3, 6, 4, 5)( 9,14,10,13)(11,16,12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,14)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 7, 8)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $8$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 9,10)(11,12)(13,16)(14,15)$ |
| $ 4, 4, 2, 2, 2, 1, 1 $ | $16$ | $4$ | $( 1, 4, 2, 3)( 5, 7)( 6, 8)( 9,12,10,11)(15,16)$ |
| $ 4, 4, 4, 2, 2 $ | $32$ | $4$ | $( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,15,10,16)(11,14)(12,13)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 7, 8)( 9,10)(11,12)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)(13,15)(14,16)$ |
| $ 4, 4, 2, 2, 2, 1, 1 $ | $16$ | $4$ | $( 1, 4, 2, 3)( 5, 7)( 6, 8)( 9,11,10,12)(13,14)$ |
| $ 4, 4, 4, 2, 2 $ | $32$ | $4$ | $( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,16,10,15)(11,13)(12,14)$ |
| $ 8, 2, 2, 2, 2 $ | $32$ | $8$ | $( 1,12, 4,10, 2,11, 3, 9)( 5,13)( 6,14)( 7,15)( 8,16)$ |
| $ 4, 4, 4, 2, 2 $ | $32$ | $4$ | $( 1,13, 3,15)( 2,14, 4,16)( 5,10)( 6, 9)( 7,12, 8,11)$ |
| $ 8, 4, 4 $ | $32$ | $8$ | $( 1,11, 3,10, 2,12, 4, 9)( 5,14, 6,13)( 7,16, 8,15)$ |
| $ 4, 4, 4, 2, 2 $ | $32$ | $4$ | $( 1,14, 4,15)( 2,13, 3,16)( 5, 9, 6,10)( 7,11)( 8,12)$ |
Group invariants
| Order: | $512=2^{9}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [512, 420058] |
| Character table: Data not available. |