Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1161$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $6$ | |
| Generators: | (1,16,5,11,3,13,8,10)(2,15,6,12,4,14,7,9), (1,16,7,11)(2,15,8,12)(3,13,5,9)(4,14,6,10), (1,6)(2,5)(3,7)(4,8)(9,14,11,15)(10,13,12,16) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 7 4: $C_4$ x 4, $C_2^2$ x 7 8: $D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$ 16: $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$ 32: $Z_8 : Z_8^\times$ x 2, $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$ 64: $((C_8 : C_2):C_2):C_2$ x 2, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T75, 16T76, 16T106 128: 16T227, 16T235, 32T1101 256: 16T633 512: 32T20345 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 8: $((C_8 : C_2):C_2):C_2$
Low degree siblings
16T1161 x 7, 16T1176 x 8, 16T1198 x 8, 16T1215 x 8, 32T35886 x 8, 32T35887 x 8, 32T35888 x 8, 32T35889 x 4, 32T35890 x 8, 32T35891 x 8, 32T35892 x 16, 32T35893 x 4, 32T36024 x 8, 32T36025 x 8, 32T36026 x 4, 32T36027 x 4, 32T36028 x 8, 32T36223 x 8, 32T36224 x 8, 32T36225 x 4, 32T36226 x 8, 32T36227 x 8, 32T36228 x 4, 32T36229 x 4, 32T36230 x 8, 32T36231 x 8, 32T36232 x 4, 32T36233 x 4, 32T36234 x 8, 32T36235 x 8, 32T36236 x 4, 32T36237 x 4, 32T36238 x 4, 32T36239 x 4, 32T36240 x 8, 32T36241 x 4, 32T36242 x 4, 32T36243 x 4, 32T36343 x 16, 32T36344 x 8, 32T36345 x 8, 32T36346 x 8, 32T36347 x 4, 32T36348 x 8, 32T36349 x 4, 32T56506 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 $ | $4$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)$ |
| $ 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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,15,12,14)(10,16,11,13)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1, 8, 3, 5)( 2, 7, 4, 6)( 9,14,12,15)(10,13,11,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 5, 6)( 7, 8)( 9,11)(10,12)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 5, 6)( 7, 8)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1,16, 5,11, 3,13, 8,10)( 2,15, 6,12, 4,14, 7, 9)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1,11, 8,16, 3,10, 5,13)( 2,12, 7,15, 4, 9, 6,14)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $32$ | $4$ | $( 3, 4)( 5, 8, 6, 7)(11,12)(13,15,14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15)(10,16)(11,14)(12,13)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1, 8, 2, 7)( 3, 6, 4, 5)( 9,14,10,13)(11,15,12,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $16$ | $2$ | $( 3, 4)( 7, 8)(13,15)(14,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12)(10,11)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $16$ | $2$ | $( 3, 4)( 7, 8)( 9,10)(11,12)(13,16)(14,15)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11)(10,12)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 5, 3, 7)( 2, 6, 4, 8)( 9,15,10,16)(11,13,12,14)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $32$ | $4$ | $( 1, 5, 4, 8)( 2, 6, 3, 7)( 9,15)(10,16)(11,13)(12,14)$ |
| $ 4, 4, 4, 4 $ | $64$ | $4$ | $( 1,16, 7,10)( 2,15, 8, 9)( 3,14, 5,11)( 4,13, 6,12)$ |
| $ 4, 4, 4, 4 $ | $64$ | $4$ | $( 1,11, 8,15)( 2,12, 7,16)( 3, 9, 6,14)( 4,10, 5,13)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $4$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,15,11,13)(10,16,12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 5, 6)( 7, 8)( 9,11)(10,12)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 9,11)(10,12)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1,16, 6,12, 4,14, 8,10)( 2,15, 5,11, 3,13, 7, 9)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1,11, 8,16, 4, 9, 6,14)( 2,12, 7,15, 3,10, 5,13)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $32$ | $4$ | $( 3, 4)( 5, 8, 6, 7)(11,12)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $32$ | $4$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15,10,16)(11,14,12,13)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $16$ | $2$ | $( 3, 4)( 7, 8)(13,16)(14,15)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12)(10,11)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $16$ | $2$ | $( 3, 4)( 7, 8)( 9,10)(11,12)(13,15)(14,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11)(10,12)(13,14)(15,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $32$ | $4$ | $( 1, 5, 3, 7)( 2, 6, 4, 8)( 9,15)(10,16)(11,13)(12,14)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 5, 4, 8)( 2, 6, 3, 7)( 9,15,10,16)(11,13,12,14)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1,16, 8, 9, 2,15, 7,10)( 3,14, 6,12, 4,13, 5,11)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1,11, 8,15, 2,12, 7,16)( 3, 9, 6,14, 4,10, 5,13)$ |
Group invariants
| Order: | $1024=2^{10}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |