Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1276$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $6$ | |
| Generators: | (1,16,2,15)(3,14,4,13)(5,6)(7,8)(9,12)(10,11), (1,6)(2,5)(3,8,4,7)(9,13,12,15,10,14,11,16), (1,10,3,11)(2,9,4,12)(5,15,7,14)(6,16,8,13) | |
| $|\Aut(F/K)|$: | $2$ |
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 6, $C_2^3$ 16: $D_{8}$ x 2, $D_4\times C_2$ x 3 32: $Z_8 : Z_8^\times$, $C_2^2 \wr C_2$, 16T29 64: $(C_4^2 : C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 16T126 128: $C_2 \wr C_2\wr C_2$ x 2, 16T409 256: 16T689 512: 16T962 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 8: $C_2 \wr C_2\wr C_2$
Low degree siblings
16T1271 x 8, 16T1276 x 15, 16T1282 x 8, 32T36786 x 8, 32T36787 x 8, 32T36788 x 8, 32T36789 x 8, 32T36790 x 4, 32T36791 x 8, 32T36792 x 16, 32T36793 x 4, 32T36794 x 8, 32T36795 x 8, 32T36796 x 8, 32T36797 x 4, 32T36798 x 8, 32T36799 x 4, 32T36800 x 8, 32T36801 x 8, 32T36802 x 4, 32T36803 x 8, 32T36804 x 4, 32T36826 x 8, 32T36827 x 8, 32T36828 x 8, 32T36829 x 8, 32T36830 x 8, 32T36831 x 8, 32T36862 x 16, 32T36863 x 8, 32T36864 x 4, 32T36865 x 8, 32T36866 x 8, 32T36867 x 4, 32T36868 x 8, 32T56379 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 1, 2)( 3, 4)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1,16, 2,15)( 3,14, 4,13)( 5, 6)( 7, 8)( 9,12)(10,11)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 1,16, 2,15)( 3,14, 4,13)( 9,11)(10,12)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,16, 3,14)( 2,15, 4,13)( 5,11, 8,10)( 6,12, 7, 9)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 5, 7)( 6, 8)( 9,12)(10,11)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 1, 4)( 2, 3)( 9,10)(11,12)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $16$ | $2$ | $( 5,12)( 6,11)( 7,10)( 8, 9)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1, 2)( 3, 4)( 5,12)( 6,11)( 7,10)( 8, 9)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $32$ | $4$ | $( 3, 4)( 7, 8)( 9,12,10,11)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $32$ | $4$ | $( 1,16, 3,13)( 2,15, 4,14)( 5, 6)(11,12)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,13, 4,15)( 2,14, 3,16)( 5, 8, 6, 7)( 9,11,10,12)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1,16, 2,15)( 3,13, 4,14)( 5,11, 6,12)( 7,10, 8, 9)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1,15, 2,16)( 3,14, 4,13)( 5,11, 6,12)( 7,10, 8, 9)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1,13)( 2,14)( 3,16)( 4,15)( 5, 9)( 6,10)( 7,12)( 8,11)$ |
| $ 8, 4, 2, 2 $ | $64$ | $8$ | $( 1, 6)( 2, 5)( 3, 8, 4, 7)( 9,13,12,15,10,14,11,16)$ |
| $ 8, 4, 2, 2 $ | $64$ | $8$ | $( 1, 8)( 2, 7)( 3, 6, 4, 5)( 9,16,11,13,10,15,12,14)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1, 5,15,11, 2, 6,16,12)( 3, 7,14,10, 4, 8,13, 9)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1, 7,15, 9, 2, 8,16,10)( 3, 5,14,12, 4, 6,13,11)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $16$ | $2$ | $( 7, 8)(11,12)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $16$ | $2$ | $( 1, 2)( 3, 4)( 7, 8)(11,12)(13,15)(14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 1, 4)( 2, 3)( 5, 7, 6, 8)( 9,11,10,12)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 3)( 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,14)(15,16)$ |
| $ 4, 4, 4, 2, 1, 1 $ | $64$ | $4$ | $( 1,16, 3,14)( 2,15, 4,13)( 5, 6)( 9,12,10,11)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $4$ | $( 5,10, 6, 9)( 7,11, 8,12)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 2)( 3, 4)( 5,10, 6, 9)( 7,11, 8,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1, 4)( 2, 3)( 5,12)( 6,11)( 7, 9)( 8,10)(13,16)(14,15)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,16, 2,15)( 3,14, 4,13)( 5,11, 7,10)( 6,12, 8, 9)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $16$ | $2$ | $( 5,10)( 6, 9)( 7,11)( 8,12)(13,14)(15,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 4)( 2, 3)( 5,12, 6,11)( 7, 9, 8,10)(13,15)(14,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 3)( 2, 4)( 5,12, 6,11)( 7, 9, 8,10)(13,16)(14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $32$ | $4$ | $( 1,16)( 2,15)( 3,14)( 4,13)( 5,11, 8, 9)( 6,12, 7,10)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $32$ | $2$ | $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,13)(10,14)(11,15)(12,16)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1, 5,15,10, 4, 8,14,11)( 2, 6,16, 9, 3, 7,13,12)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1, 7,16,11, 4, 6,13,10)( 2, 8,15,12, 3, 5,14, 9)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,12, 3,10)( 2,11, 4, 9)( 5,15, 8,14)( 6,16, 7,13)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,12, 4, 9)( 2,11, 3,10)( 5,16, 7,14)( 6,15, 8,13)$ |
Group invariants
| Order: | $1024=2^{10}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |