Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1113$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $6$ | |
| Generators: | (1,14,2,13)(3,16)(4,15)(5,10,7,12)(6,9,8,11), (1,4,2,3)(5,14,8,15,6,13,7,16)(9,10)(11,12), (1,15,12,6,3,14,9,7,2,16,11,5,4,13,10,8) | |
| $|\Aut(F/K)|$: | $2$ |
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 12, $C_4\times C_2$ x 6, $C_2^3$ 16: $D_4\times C_2$ x 6, $C_2^2:C_4$ x 12, $C_4\times C_2^2$ 32: $C_2^2 \wr C_2$ x 4, $C_2^3 : C_4 $ x 4, $C_2 \times (C_2^2:C_4)$ x 3 64: $((C_8 : C_2):C_2):C_2$ x 4, 16T76 x 2, 16T79 128: 16T227 x 2, 16T240 256: 16T502 512: 32T23372 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$
Low degree siblings
16T1113 x 15, 32T35519 x 8, 32T35520 x 16, 32T35521 x 32, 32T35522 x 8, 32T35523 x 16, 32T35524 x 16, 32T35525 x 8, 32T35526 x 16, 32T48765 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 $ | $2$ | $2$ | $( 5, 6)( 7, 8)(13,14)(15,16)$ |
| $ 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, 1, 1, 1, 1 $ | $16$ | $2$ | $( 1, 2)( 5, 7)( 6, 8)( 9,11)(10,12)(13,14)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 1, 2)( 3, 4)( 5, 7, 6, 8)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)$ |
| $ 4, 4, 4, 2, 2 $ | $64$ | $4$ | $( 1,14, 2,13)( 3,16)( 4,15)( 5,10, 7,12)( 6, 9, 8,11)$ |
| $ 4, 4, 4, 2, 2 $ | $64$ | $4$ | $( 1,13, 2,14)( 3,16)( 4,15)( 5,12, 7,10)( 6,11, 8, 9)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1,10, 4,11, 2, 9, 3,12)( 5,14, 8,16, 6,13, 7,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1,12, 2,11)( 3,10, 4, 9)( 5,13)( 6,14)( 7,15)( 8,16)$ |
| $ 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 $ | $16$ | $2$ | $( 1, 2)( 5, 7)( 6, 8)( 9,12)(10,11)(15,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,10)(11,12)(13,16,14,15)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 5, 7, 6, 8)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1,10, 3,12, 2, 9, 4,11)( 5,14, 7,15, 6,13, 8,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,13, 6,14)( 7,15, 8,16)$ |
| $ 8, 4, 1, 1, 1, 1 $ | $16$ | $8$ | $( 1, 9, 4,11, 2,10, 3,12)(13,16,14,15)$ |
| $ 8, 4, 2, 2 $ | $16$ | $8$ | $( 1, 9, 4,11, 2,10, 3,12)( 5, 6)( 7, 8)(13,15,14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1,11)( 2,12)( 3,10)( 4, 9)( 5, 7)( 6, 8)(13,16)(14,15)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 1,11, 2,12)( 3,10, 4, 9)( 7, 8)(13,14)$ |
| $ 16 $ | $64$ | $16$ | $( 1, 8,11,13, 3, 5,10,16, 2, 7,12,14, 4, 6, 9,15)$ |
| $ 16 $ | $64$ | $16$ | $( 1, 6,11,14, 4, 8, 9,15, 2, 5,12,13, 3, 7,10,16)$ |
| $ 8, 4, 1, 1, 1, 1 $ | $16$ | $8$ | $( 1,10, 3,11, 2, 9, 4,12)(13,15,14,16)$ |
| $ 8, 4, 2, 2 $ | $16$ | $8$ | $( 1,10, 3,11, 2, 9, 4,12)( 5, 6)( 7, 8)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1,12, 2,11)( 3, 9, 4,10)( 5, 7)( 6, 8)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $16$ | $2$ | $( 1,12)( 2,11)( 3, 9)( 4,10)( 7, 8)(15,16)$ |
| $ 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 $ | $4$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $32$ | $2$ | $( 1, 2)( 5, 7)( 6, 8)( 9,11)(10,12)(15,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 1, 2)( 3, 4)( 5, 7, 6, 8)(13,16,14,15)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 1, 3, 2, 4)( 9,12,10,11)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 3, 2, 4)( 5, 6)( 7, 8)( 9,12,10,11)(13,14)(15,16)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$ |
| $ 4, 4, 4, 2, 2 $ | $64$ | $4$ | $( 1,14)( 2,13)( 3,16, 4,15)( 5,10, 7,12)( 6, 9, 8,11)$ |
| $ 4, 4, 4, 2, 2 $ | $64$ | $4$ | $( 1,13)( 2,14)( 3,16, 4,15)( 5,12, 7,10)( 6,11, 8, 9)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1,10, 4,11, 2, 9, 3,12)( 5,14, 7,15, 6,13, 8,16)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,12, 2,11)( 3,10, 4, 9)( 5,13, 6,14)( 7,15, 8,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,16)( 6,15)( 7,13)( 8,14)$ |
| $ 8, 4, 1, 1, 1, 1 $ | $16$ | $8$ | $( 1, 9, 4,11, 2,10, 3,12)(13,15,14,16)$ |
| $ 8, 4, 2, 2 $ | $16$ | $8$ | $( 1, 9, 4,11, 2,10, 3,12)( 5, 6)( 7, 8)(13,16,14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $16$ | $2$ | $( 1,11)( 2,12)( 3,10)( 4, 9)( 5, 7)( 6, 8)(13,15)(14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $16$ | $4$ | $( 1,11, 2,12)( 3,10, 4, 9)( 7, 8)(15,16)$ |
| $ 16 $ | $64$ | $16$ | $( 1, 8,11,13, 4, 6, 9,15, 2, 7,12,14, 3, 5,10,16)$ |
| $ 16 $ | $64$ | $16$ | $( 1, 6,11,14, 3, 7,10,16, 2, 5,12,13, 4, 8, 9,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 3)( 2, 4)( 5,14, 6,13)( 7,15, 8,16)( 9,11)(10,12)$ |
| $ 8, 4, 1, 1, 1, 1 $ | $16$ | $8$ | $( 1, 3, 2, 4)( 5,13, 7,15, 6,14, 8,16)$ |
| $ 8, 4, 2, 2 $ | $16$ | $8$ | $( 1, 4, 2, 3)( 5,13, 7,15, 6,14, 8,16)( 9,10)(11,12)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $16$ | $2$ | $( 1, 2)( 5,13)( 6,14)( 7,16)( 8,15)(11,12)$ |
Group invariants
| Order: | $1024=2^{10}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |