Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $869$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $4$ | |
| Generators: | (3,4)(5,7)(6,8)(9,11)(10,12)(13,14), (1,5,2,6)(3,7)(4,8)(9,16,10,15)(11,13)(12,14), (1,16,2,15)(3,13)(4,14)(5,11)(6,12)(7,9,8,10) | |
| $|\Aut(F/K)|$: | $2$ |
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: 16T208, 16T218, 16T230 256: 32T3729 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 8: $D_4\times C_2$
Low degree siblings
16T847 x 8, 16T869 x 3, 16T884 x 4, 32T10140 x 8, 32T10141 x 4, 32T10142 x 4, 32T10296 x 4, 32T10297 x 4, 32T10298 x 8, 32T10299 x 2, 32T10300 x 4, 32T10301 x 4, 32T10302 x 2, 32T10380 x 4, 32T10381 x 8, 32T10382 x 2, 32T10383 x 4, 32T10384 x 4, 32T10385 x 2, 32T10386 x 4, 32T19784 x 2, 32T19866 x 2, 32T20111 x 2, 32T20142 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, 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 $ | $16$ | $2$ | $( 3, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,14)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 5, 6)( 7, 8)$ |
| $ 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 5, 6)( 7, 8)( 9,10)(11,12)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $16$ | $2$ | $( 1, 3)( 2, 4)( 5, 6)(11,12)(13,15)(14,16)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$ |
| $ 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $4$ | $( 5, 7, 6, 8)( 9,11,10,12)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $8$ | $4$ | $( 1, 2)( 3, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 3, 4)( 7, 8)(11,12)(13,14)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 4)( 5, 6)(11,12)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 1, 4, 2, 3)( 9,10)(11,12)(13,15,14,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $8$ | $4$ | $( 1, 3, 2, 4)( 5, 6)( 7, 8)(13,16,14,15)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 3, 4)( 5, 6)(11,12)(15,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 5, 2, 6)( 3, 7)( 4, 8)( 9,16,10,15)(11,13)(12,14)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 7, 4, 6)( 2, 8, 3, 5)( 9,16,12,13)(10,15,11,14)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 6)( 2, 5)( 3, 8, 4, 7)( 9,16,10,15)(11,13)(12,14)$ |
| $ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 7, 3, 6)( 2, 8, 4, 5)( 9,16,12,14)(10,15,11,13)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 5, 2, 6)( 3, 8)( 4, 7)( 9,16,10,15)(11,14)(12,13)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1, 6)( 2, 5)( 3, 7, 4, 8)( 9,16,10,15)(11,14)(12,13)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $32$ | $4$ | $( 1,16, 2,15)( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9, 8,10)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1,16, 2,15)( 3,14)( 4,13)( 5, 9, 6,10)( 7,11)( 8,12)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1,14, 2,13)( 3,15)( 4,16)( 5,12, 6,11)( 7, 9)( 8,10)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1,16, 2,15)( 3,13)( 4,14)( 5,10, 6, 9)( 7,11)( 8,12)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $32$ | $4$ | $( 1,16, 2,15)( 3,14)( 4,13)( 5,12, 6,11)( 7, 9)( 8,10)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1,13)( 2,14)( 3,15, 4,16)( 5,12)( 6,11)( 7,10, 8, 9)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1,10, 3,11, 2, 9, 4,12)( 5,13, 7,16, 6,14, 8,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1,12)( 2,11)( 3, 9)( 4,10)( 5,14, 6,13)( 7,16, 8,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1,12)( 2,11)( 3, 9)( 4,10)( 5,14)( 6,13)( 7,16)( 8,15)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1,11, 2,12)( 3,10, 4, 9)( 5,14, 6,13)( 7,16, 8,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $16$ | $4$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,13, 6,14)( 7,16, 8,15)$ |
| $ 8, 8 $ | $32$ | $8$ | $( 1,10, 4,11, 2, 9, 3,12)( 5,14, 7,16, 6,13, 8,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,13)( 6,14)( 7,16)( 8,15)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1,11, 2,12)( 3, 9, 4,10)( 5,13, 6,14)( 7,16, 8,15)$ |
Group invariants
| Order: | $512=2^{9}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [512, 46955] |
| Character table: Data not available. |