Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1262$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $6$ | |
| Generators: | (5,10,6,9)(7,11,8,12), (1,14,3,15,2,13,4,16)(5,7)(6,8)(9,11)(10,12), (1,6,13,9,2,5,14,10)(3,8,15,12,4,7,16,11) | |
| $|\Aut(F/K)|$: | $2$ |
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: 16T659, 16T662, 16T689 512: 32T16602 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
16T1262 x 7, 32T36715 x 4, 32T36716 x 4, 32T36717 x 4, 32T36718 x 4, 32T36719 x 4, 32T36720 x 4, 32T36721 x 4, 32T56400 x 4, 32T67670 x 2, 32T67777 x 2, 32T68446 x 2Siblings 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 $ | $2$ | $2$ | $( 5, 6)( 7, 8)( 9,10)(11,12)$ |
| $ 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $4$ | $( 5,10, 6, 9)( 7,11, 8,12)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 2)( 3, 4)( 5,10, 6, 9)( 7,11, 8,12)(13,14)(15,16)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,13, 2,14)( 3,15, 4,16)( 5,10, 6, 9)( 7,11, 8,12)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 1, 4, 2, 3)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 4, 2, 3)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1, 4, 2, 3)( 5,10, 6, 9)( 7,11, 8,12)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)$ |
| $ 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, 1, 1, 1, 1 $ | $16$ | $2$ | $( 5,10)( 6, 9)( 7,11)( 8,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1,13)( 2,14)( 3,15)( 4,16)( 5,10)( 6, 9)( 7,11)( 8,12)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 1, 4, 2, 3)( 9,10)(11,12)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 4, 2, 3)( 5,10)( 6, 9)( 7,11)( 8,12)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $32$ | $2$ | $( 3, 4)( 7, 8)(11,12)(13,16)(14,15)$ |
| $ 4, 4, 2, 2, 2, 1, 1 $ | $32$ | $4$ | $( 3, 4)( 5,10, 6, 9)( 7,12, 8,11)(13,16)(14,15)$ |
| $ 8, 2, 2, 1, 1, 1, 1 $ | $32$ | $8$ | $( 1,13, 3,16, 2,14, 4,15)( 7, 8)(11,12)$ |
| $ 8, 4, 4 $ | $16$ | $8$ | $( 1,13, 3,16, 2,14, 4,15)( 5,10, 6, 9)( 7,12, 8,11)$ |
| $ 8, 4, 4 $ | $16$ | $8$ | $( 1,13, 3,16, 2,14, 4,15)( 5, 9, 6,10)( 7,11, 8,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $32$ | $2$ | $( 3, 4)( 5,11)( 6,12)( 7,10)( 8, 9)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $32$ | $2$ | $( 3, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)$ |
| $ 8, 2, 2, 2, 2 $ | $32$ | $8$ | $( 1,13, 3,16, 2,14, 4,15)( 5,11)( 6,12)( 7,10)( 8, 9)$ |
| $ 8, 2, 2, 2, 2 $ | $32$ | $8$ | $( 1,13, 3,16, 2,14, 4,15)( 5, 8)( 6, 7)( 9,12)(10,11)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $4$ | $( 9,12,10,11)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 5, 6)( 7, 8)( 9,11,10,12)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11,10,12)(13,15,14,16)$ |
| $ 8, 4, 1, 1, 1, 1 $ | $32$ | $8$ | $( 5,10, 8,12, 6, 9, 7,11)(13,16,14,15)$ |
| $ 8, 4, 2, 2 $ | $32$ | $8$ | $( 1, 2)( 3, 4)( 5,10, 8,12, 6, 9, 7,11)(13,15,14,16)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1,13, 3,15, 2,14, 4,16)( 5,10, 8,12, 6, 9, 7,11)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1, 6,13, 9, 2, 5,14,10)( 3, 8,15,12, 4, 7,16,11)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 9, 2,10)( 3,12, 4,11)( 5,14, 6,13)( 7,16, 8,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $32$ | $2$ | $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,14)( 6,13)( 7,16)( 8,15)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1, 6, 3, 8, 2, 5, 4, 7)( 9,16,11,14,10,15,12,13)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1, 9,16, 8, 2,10,15, 7)( 3,12,13, 5, 4,11,14, 6)$ |
| $ 16 $ | $64$ | $16$ | $( 1, 6,16,12, 3, 7,13, 9, 2, 5,15,11, 4, 8,14,10)$ |
| $ 8, 4, 4 $ | $32$ | $8$ | $( 1, 9, 2,10)( 3,11, 4,12)( 5,15, 8,14, 6,16, 7,13)$ |
| $ 8, 2, 2, 2, 2 $ | $32$ | $8$ | $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,15, 7,13, 6,16, 8,14)$ |
| $ 16 $ | $64$ | $16$ | $( 1, 6,15,12, 3, 7,14, 9, 2, 5,16,11, 4, 8,13,10)$ |
| $ 8, 4, 4 $ | $32$ | $8$ | $( 1, 9, 2,10)( 3,11, 4,12)( 5,16, 8,13, 6,15, 7,14)$ |
| $ 8, 2, 2, 2, 2 $ | $32$ | $8$ | $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,16, 7,14, 6,15, 8,13)$ |
Group invariants
| Order: | $1024=2^{10}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |