Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1651$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,5,3,7,2,6,4,8)(9,11,10,12)(13,16,14,15), (1,15,7,13)(2,16,8,14)(3,11)(4,12)(5,10,6,9) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 8: $D_{4}$ 72: $C_3^2:D_4$ 1152: $S_4\wr C_2$ 2304: 32T205478 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Degree 8: $S_4\wr C_2$
Low degree siblings
16T1651 x 3, 32T396914 x 2, 32T396915 x 2, 32T396916 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
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 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)$ |
| $ 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 $ | $12$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)$ |
| $ 4, 4, 4, 4 $ | $36$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,13,10,14)(11,15,12,16)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $16$ | $3$ | $( 3, 5, 8)( 4, 6, 7)$ |
| $ 6, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $16$ | $6$ | $( 1, 2)( 3, 6, 8, 4, 5, 7)$ |
| $ 3, 3, 2, 2, 2, 2, 1, 1 $ | $16$ | $6$ | $( 3, 5, 8)( 4, 6, 7)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 6, 2, 2, 2, 2, 2 $ | $16$ | $6$ | $( 1, 2)( 3, 6, 8, 4, 5, 7)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 3, 3, 1, 1 $ | $96$ | $12$ | $( 3, 5, 8)( 4, 6, 7)( 9,14,10,13)(11,16,12,15)$ |
| $ 6, 4, 4, 2 $ | $96$ | $12$ | $( 1, 2)( 3, 6, 8, 4, 5, 7)( 9,14,10,13)(11,16,12,15)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1 $ | $64$ | $3$ | $( 3, 5, 8)( 4, 6, 7)(11,13,15)(12,14,16)$ |
| $ 6, 3, 3, 2, 1, 1 $ | $128$ | $6$ | $( 1, 2)( 3, 6, 8, 4, 5, 7)(11,13,15)(12,14,16)$ |
| $ 6, 6, 2, 2 $ | $64$ | $6$ | $( 1, 2)( 3, 6, 8, 4, 5, 7)( 9,10)(11,14,15,12,13,16)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $2$ | $( 3, 4)( 5, 7)( 6, 8)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $24$ | $2$ | $( 3, 4)( 5, 7)( 6, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 4, 4, 2, 2, 2, 1, 1 $ | $144$ | $4$ | $( 3, 4)( 5, 7)( 6, 8)( 9,14,10,13)(11,16,12,15)$ |
| $ 8, 2, 2, 2, 2 $ | $12$ | $8$ | $( 1, 5, 3, 7, 2, 6, 4, 8)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 8, 2, 2, 2, 2 $ | $12$ | $8$ | $( 1, 6, 3, 8, 2, 5, 4, 7)( 9,10)(11,12)(13,14)(15,16)$ |
| $ 8, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $8$ | $( 1, 5, 3, 7, 2, 6, 4, 8)$ |
| $ 8, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $8$ | $( 1, 6, 3, 8, 2, 5, 4, 7)$ |
| $ 8, 4, 4 $ | $72$ | $8$ | $( 1, 5, 3, 7, 2, 6, 4, 8)( 9,13,10,14)(11,15,12,16)$ |
| $ 8, 4, 4 $ | $72$ | $8$ | $( 1, 6, 3, 8, 2, 5, 4, 7)( 9,13,10,14)(11,15,12,16)$ |
| $ 3, 3, 2, 2, 2, 1, 1, 1, 1 $ | $192$ | $6$ | $( 3, 4)( 5, 7)( 6, 8)(11,13,15)(12,14,16)$ |
| $ 6, 2, 2, 2, 2, 1, 1 $ | $192$ | $6$ | $( 3, 4)( 5, 7)( 6, 8)( 9,10)(11,14,15,12,13,16)$ |
| $ 8, 6, 2 $ | $96$ | $24$ | $( 1, 5, 3, 7, 2, 6, 4, 8)( 9,10)(11,14,15,12,13,16)$ |
| $ 8, 6, 2 $ | $96$ | $24$ | $( 1, 6, 3, 8, 2, 5, 4, 7)( 9,10)(11,14,15,12,13,16)$ |
| $ 8, 3, 3, 1, 1 $ | $96$ | $24$ | $( 1, 5, 3, 7, 2, 6, 4, 8)(11,13,15)(12,14,16)$ |
| $ 8, 3, 3, 1, 1 $ | $96$ | $24$ | $( 1, 6, 3, 8, 2, 5, 4, 7)(11,13,15)(12,14,16)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $144$ | $2$ | $( 3, 4)( 5, 7)( 6, 8)(11,16)(12,15)(13,14)$ |
| $ 8, 2, 2, 2, 1, 1 $ | $144$ | $8$ | $( 1, 5, 3, 7, 2, 6, 4, 8)( 9,10)(11,15)(12,16)$ |
| $ 8, 2, 2, 2, 1, 1 $ | $144$ | $8$ | $( 1, 6, 3, 8, 2, 5, 4, 7)( 9,10)(11,15)(12,16)$ |
| $ 8, 8 $ | $72$ | $8$ | $( 1, 5, 3, 7, 2, 6, 4, 8)( 9,16,13,11,10,15,14,12)$ |
| $ 8, 8 $ | $36$ | $8$ | $( 1, 6, 3, 8, 2, 5, 4, 7)( 9,16,13,11,10,15,14,12)$ |
| $ 8, 8 $ | $36$ | $8$ | $( 1, 5, 3, 7, 2, 6, 4, 8)( 9,15,13,12,10,16,14,11)$ |
| $ 4, 4, 4, 2, 2 $ | $576$ | $4$ | $( 1,15, 7,13)( 2,16, 8,14)( 3,11)( 4,12)( 5,10, 6, 9)$ |
| $ 16 $ | $288$ | $16$ | $( 1,16, 5, 9, 7,14, 4,11, 2,15, 6,10, 8,13, 3,12)$ |
| $ 16 $ | $288$ | $16$ | $( 1,16, 6,10, 7,14, 3,12, 2,15, 5, 9, 8,13, 4,11)$ |
| $ 12, 4 $ | $384$ | $12$ | $( 1,15, 5,10, 8,14, 2,16, 6, 9, 7,13)( 3,11, 4,12)$ |
| $ 6, 6, 2, 2 $ | $384$ | $6$ | $( 1,15, 6, 9, 8,14)( 2,16, 5,10, 7,13)( 3,11)( 4,12)$ |
| $ 8, 8 $ | $288$ | $8$ | $( 1,15, 8,14, 2,16, 7,13)( 3,11, 6, 9, 4,12, 5,10)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $48$ | $2$ | $( 1,15)( 2,16)( 3,11)( 4,12)( 5,10)( 6, 9)( 7,13)( 8,14)$ |
| $ 4, 4, 4, 4 $ | $48$ | $4$ | $( 1,15, 2,16)( 3,11, 4,12)( 5,10, 6, 9)( 7,13, 8,14)$ |
Group invariants
| Order: | $4608=2^{9} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |