Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1186$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $6$ | |
| Generators: | (1,8)(2,7)(3,6)(4,5)(9,13,12,16,10,14,11,15), (1,16,8,10,4,14,6,12,2,15,7,9,3,13,5,11), (3,4)(5,8)(6,7)(11,12)(15,16) | |
| $|\Aut(F/K)|$: | $2$ |
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: 16T483, 16T518, 16T633 512: 32T12364 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
16T1186 x 15, 32T36116 x 8, 32T36117 x 32, 32T36118 x 16, 32T36119 x 8, 32T36120 x 16, 32T36121 x 8, 32T36122 x 16, 32T36123 x 16, 32T56505 x 8Siblings 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 $ | $2$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ |
| $ 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 9,12,10,11)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,10,11)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 5, 6)( 7, 8)(13,14)(15,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 5, 6)( 7, 8)( 9,12,10,11)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$ |
| $ 8, 2, 2, 2, 2 $ | $32$ | $8$ | $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,13,12,16,10,14,11,15)$ |
| $ 8, 4, 4 $ | $32$ | $8$ | $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,13,12,16,10,14,11,15)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $4$ | $( 5, 7, 6, 8)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 5, 7, 6, 8)( 9,10)(11,12)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 1, 2)( 3, 4)( 5, 8, 6, 7)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 2)( 3, 4)( 5, 8, 6, 7)( 9,10)(11,12)(13,16,14,15)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $32$ | $2$ | $( 3, 4)( 7, 8)(11,12)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $32$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)(11,12)(13,16)(14,15)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,13,10,14)(11,16,12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,14)(10,13)(11,15)(12,16)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 8, 3, 5, 2, 7, 4, 6)( 9,13,12,15,10,14,11,16)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 8, 3, 5, 2, 7, 4, 6)( 9,16,12,13,10,15,11,14)$ |
| $ 16 $ | $64$ | $16$ | $( 1,16, 8,10, 4,14, 6,12, 2,15, 7, 9, 3,13, 5,11)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1,10, 5,15, 2, 9, 6,16)( 3,12, 7,13, 4,11, 8,14)$ |
| $ 8, 8 $ | $64$ | $8$ | $( 1,13, 5,12, 2,14, 6,11)( 3,16, 7, 9, 4,15, 8,10)$ |
| $ 16 $ | $64$ | $16$ | $( 1,12, 7,13, 4, 9, 5,16, 2,11, 8,14, 3,10, 6,15)$ |
| $ 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, 1, 1, 1, 1 $ | $4$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 9,12,10,11)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,10,11)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)(13,14)(15,16)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)$ |
| $ 8, 2, 2, 2, 2 $ | $32$ | $8$ | $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,13,11,15,10,14,12,16)$ |
| $ 8, 4, 4 $ | $32$ | $8$ | $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,13,11,15,10,14,12,16)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $4$ | $( 5, 7, 6, 8)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 5, 7, 6, 8)( 9,10)(11,12)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 1, 2)( 3, 4)( 5, 8, 6, 7)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 2)( 3, 4)( 5, 8, 6, 7)( 9,10)(11,12)(13,15,14,16)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $32$ | $2$ | $( 3, 4)( 7, 8)(11,12)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $32$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)(11,12)(13,15)(14,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $32$ | $4$ | $( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1, 8, 3, 5, 2, 7, 4, 6)( 9,13,11,16,10,14,12,15)$ |
| $ 16 $ | $64$ | $16$ | $( 1,16, 7, 9, 3,13, 6,12, 2,15, 8,10, 4,14, 5,11)$ |
| $ 4, 4, 4, 4 $ | $64$ | $4$ | $( 1,10, 5,15)( 2, 9, 6,16)( 3,12, 7,13)( 4,11, 8,14)$ |
| $ 4, 4, 4, 4 $ | $64$ | $4$ | $( 1,13, 6,11)( 2,14, 5,12)( 3,16, 8,10)( 4,15, 7, 9)$ |
| $ 16 $ | $64$ | $16$ | $( 1,12, 7,13, 3,10, 6,15, 2,11, 8,14, 4, 9, 5,16)$ |
Group invariants
| Order: | $1024=2^{10}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |