Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $511$ | |
| Group : | $C_2^2.C_2^5.C_2$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $3$ | |
| Generators: | (3,7)(4,8)(9,15)(10,16)(11,14)(12,13), (1,9,2,10)(3,15,4,16)(5,14,6,13)(7,11,8,12), (1,7)(2,8)(3,5)(4,6)(9,15)(10,16)(11,14)(12,13), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,16)(12,15) | |
| $|\Aut(F/K)|$: | $4$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 15 4: $C_2^2$ x 35 8: $D_{4}$ x 20, $C_2^3$ x 15 16: $D_4\times C_2$ x 30, $C_2^4$ 32: $C_2^2 \wr C_2$ x 8, $C_2^3 : D_4 $ x 2, $C_2^2 \times D_4$ x 5 64: $(C_4^2 : C_2):C_2$ x 4, 16T87, 16T105 x 2, 16T109 x 4 128: 16T265 x 2, 32T1237 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$ x 3
Degree 8: $C_2^2 \wr C_2$, $(C_4^2 : C_2):C_2$ x 2
Low degree siblings
16T511 x 63, 32T2537 x 16, 32T2538 x 32, 32T2539 x 16, 32T2540 x 32, 32T2541 x 16, 32T6814 x 8, 32T7024 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 $ | $2$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 9,11,14,15)(10,12,13,16)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $4$ | $( 9,12,14,16)(10,11,13,15)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 9,13)(10,14)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 9,14)(10,13)(11,15)(12,16)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 3, 7)( 4, 8)(11,15)(12,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 7)( 4, 8)( 9,10)(11,16)(12,15)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 7)( 4, 8)( 9,11)(10,12)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 7)( 4, 8)( 9,12)(10,11)(13,15)(14,16)$ |
| $ 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, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11,14,15)(10,12,13,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,14,16)(10,11,13,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14)(10,13)(11,15)(12,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,10)(11,16)(12,15)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,11)(10,12)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,12)(10,11)(13,15)(14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$ |
| $ 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, 4, 4 $ | $2$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,11,14,15)(10,12,13,16)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,12,14,16)(10,11,13,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,14)(10,13)(11,15)(12,16)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,15,14,11)(10,16,13,12)$ |
| $ 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,16,14,12)(10,15,13,11)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,12,14,16)(10,11,13,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $4$ | $4$ | $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,14)(10,13)(11,15)(12,16)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,16,14,12)(10,15,13,11)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,15)(12,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,13)(10,14)(11,16)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,14)( 6,13)( 7,15)( 8,16)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 9, 2,10)( 3,11, 4,12)( 5,14, 6,13)( 7,15, 8,16)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 9, 3,11, 5,14, 7,15)( 2,10, 4,12, 6,13, 8,16)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 9, 4,12, 5,14, 8,16)( 2,10, 3,11, 6,13, 7,15)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 9, 5,14)( 2,10, 6,13)( 3,11, 7,15)( 4,12, 8,16)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 9, 6,13)( 2,10, 5,14)( 3,11, 8,16)( 4,12, 7,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 9)( 2,10)( 3,15)( 4,16)( 5,14)( 6,13)( 7,11)( 8,12)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 9, 2,10)( 3,15, 4,16)( 5,14, 6,13)( 7,11, 8,12)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 9, 3,15, 5,14, 7,11)( 2,10, 4,16, 6,13, 8,12)$ |
| $ 8, 8 $ | $16$ | $8$ | $( 1, 9, 4,16, 5,14, 8,12)( 2,10, 3,15, 6,13, 7,11)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 9, 5,14)( 2,10, 6,13)( 3,15, 7,11)( 4,16, 8,12)$ |
| $ 4, 4, 4, 4 $ | $8$ | $4$ | $( 1, 9, 6,13)( 2,10, 5,14)( 3,15, 8,12)( 4,16, 7,11)$ |
Group invariants
| Order: | $256=2^{8}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [256, 16894] |
| Character table: Data not available. |