Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1025$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $4$ | |
| Generators: | (1,13,2,14)(3,12,4,11)(5,9,6,10)(7,16,8,15), (1,8,2,7)(3,5,4,6), (1,2)(5,6)(9,13)(10,14) | |
| $|\Aut(F/K)|$: | $4$ |
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: $(C_4^2 : C_2):C_2$ x 2, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 32T320 128: $C_2 \wr C_2\wr C_2$ x 2, 16T336, 16T342, 16T350, 16T382, 16T408 256: 32T5721, 32T5807, 32T6155 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 8: $(C_4^2 : C_2):C_2$, $C_2 \wr C_2\wr C_2$ x 2
Low degree siblings
16T1025 x 31, 32T11148 x 8, 32T11149 x 8, 32T11150 x 8, 32T11151 x 8, 32T11152 x 8, 32T11153 x 8, 32T11154 x 8, 32T20622 x 4, 32T20632 x 4, 32T20633 x 4, 32T20639 x 4, 32T20964 x 4, 32T20966 x 4, 32T21008 x 4, 32T21074 x 8, 32T21484 x 4, 32T21497 x 4, 32T21501 x 4, 32T21506 x 4, 32T21574 x 4, 32T21631 x 4, 32T21847 x 4, 32T21848 x 4, 32T21877 x 4, 32T21878 x 4, 32T21880 x 4Siblings 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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 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, 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 $ | $16$ | $4$ | $( 1,13, 2,14)( 3,12, 4,11)( 5, 9, 6,10)( 7,16, 8,15)$ |
| $ 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 $ | $4$ | $2$ | $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,10)(11,15)(12,16)(13,14)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,12,10,11)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,11, 5,15)( 2,12, 6,16)( 3,10, 7,14)( 4, 9, 8,13)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,15,13,11)(10,16,14,12)$ |
| $ 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 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,13)(10,14)(11,15)(12,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ |
| $ 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 $ | $16$ | $2$ | $( 1,13)( 2,14)( 3,12)( 4,11)( 5, 9)( 6,10)( 7,16)( 8,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 8)( 4, 7)( 9,10)(11,15)(12,16)(13,14)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,11,10,12)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,12,10,11)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,11, 6,16)( 2,12, 5,15)( 3,10, 8,13)( 4, 9, 7,14)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,16,13,12)(10,15,14,11)$ |
| $ 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, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14)(10,13)(11,16)(12,15)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $4$ | $( 9,11,10,12)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,10,11)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15,10,16)(11,14,12,13)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,16,10,15)(11,13,12,14)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1,13, 7,16, 2,14, 8,15)( 3,12, 6,10, 4,11, 5, 9)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 3, 8)( 4, 7)( 9,16,13,12)(10,15,14,11)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,15,13,11)(10,16,14,12)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1,13, 4,16, 2,14, 3,15)( 5, 9, 8,12, 6,10, 7,11)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 4)( 7, 8)( 9,14)(10,13)(11,12)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 1, 2)( 5, 6)( 9,13)(10,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,10)(11,16)(12,15)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 1, 6)( 2, 5)( 3, 7)( 4, 8)(11,15)(12,16)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,13, 5, 9)( 2,14, 6,10)( 3,11, 4,12)( 7,15, 8,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $32$ | $4$ | $( 1,16)( 2,15)( 3, 9, 8,14)( 4,10, 7,13)( 5,12)( 6,11)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,15,14,12)(10,16,13,11)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,16,14,11)(10,15,13,12)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 3, 4)( 7, 8)( 9,16,14,11)(10,15,13,12)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,12,14,15)(10,11,13,16)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1,13, 4,12, 5, 9, 8,16)( 2,14, 3,11, 6,10, 7,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $16$ | $2$ | $( 3, 7)( 4, 8)( 9,11)(10,12)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,12)(10,11)(13,16)(14,15)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1,13, 7,12, 5, 9, 3,16)( 2,14, 8,11, 6,10, 4,15)$ |
Group invariants
| Order: | $512=2^{9}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [512, 49732] |
| Character table: Data not available. |