Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $864$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $5$ | |
| Generators: | (1,12,5,16)(2,11,6,15)(3,9,7,13)(4,10,8,14), (1,3,2,4)(5,6)(7,8)(9,12,10,11), (1,6,4,8,2,5,3,7)(9,14,12,16,10,13,11,15) | |
| $|\Aut(F/K)|$: | $2$ |
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 8, $C_4\times C_2$ x 6, $C_2^3$ 16: $D_4\times C_2$ x 4, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$ 32: $C_2^2 \wr C_2$, $C_2^3 : C_4 $ x 4, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37 64: $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T76 x 2, 32T239 128: 16T230, 16T234, 16T235 256: 32T4050 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 8: $C_2^3: C_4$
Low degree siblings
16T864 x 7, 32T10272 x 4, 32T10273 x 8, 32T10274 x 4, 32T10275 x 8, 32T10276 x 4, 32T20115 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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,12, 5,16)( 2,11, 6,15)( 3, 9, 7,13)( 4,10, 8,14)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,16, 5,12)( 2,15, 6,11)( 3,13, 7, 9)( 4,14, 8,10)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 9,11,10,12)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,10,11)(13,15,14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15)(10,16)(11,14)(12,13)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 5, 6)( 7, 8)(13,14)(15,16)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,13,10,14)(11,15,12,16)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,12, 5,15)( 2,11, 6,16)( 3, 9, 7,14)( 4,10, 8,13)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,16, 6,12)( 2,15, 5,11)( 3,13, 8, 9)( 4,14, 7,10)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 5, 6)( 7, 8)( 9,11,10,12)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,15,10,16)(11,14,12,13)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $4$ | $( 5, 8, 6, 7)(13,16,14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,10)(11,12)(13,15,14,16)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1, 5, 4, 8, 2, 6, 3, 7)( 9,13,12,16,10,14,11,15)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 5, 8, 6, 7)( 9,10)(11,12)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 1, 2)( 3, 4)( 5, 7, 6, 8)(13,16,14,15)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1,12, 5,14, 2,11, 6,13)( 3, 9, 7,16, 4,10, 8,15)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1,16, 8,11, 2,15, 7,12)( 3,13, 5,10, 4,14, 6, 9)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 4)( 7, 8)(11,12)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,15,10,16)(11,13,12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,14)(10,13)(11,15)(12,16)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,12, 6,15)( 2,11, 5,16)( 3,10, 8,13)( 4, 9, 7,14)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,16, 6,11)( 2,15, 5,12)( 3,14, 8, 9)( 4,13, 7,10)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $16$ | $2$ | $( 3, 4)( 7, 8)( 9,11)(10,12)(13,16)(14,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,15,10,16)(11,13,12,14)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,12, 8,13)( 2,11, 7,14)( 3,10, 5,16)( 4, 9, 6,15)$ |
| $ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,16, 7,10)( 2,15, 8, 9)( 3,14, 6,11)( 4,13, 5,12)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $32$ | $2$ | $( 3, 4)( 5, 8)( 6, 7)(11,12)(13,16)(14,15)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 5, 4, 7, 2, 6, 3, 8)( 9,13,12,15,10,14,11,16)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 5, 4, 7, 2, 6, 3, 8)( 9,14,12,16,10,13,11,15)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,12, 6,13)( 2,11, 5,14)( 3,10, 8,16)( 4, 9, 7,15)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,16, 7,11)( 2,15, 8,12)( 3,14, 6, 9)( 4,13, 5,10)$ |
Group invariants
| Order: | $512=2^{9}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [512, 59944] |
| Character table: Data not available. |