Group action invariants
| Degree $n$ : | $16$ | |
| Transitive number $t$ : | $1535$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,16,2,15)(3,13,4,14)(5,6)(7,8), (1,3,2,4)(5,12,6,11)(7,10,8,9)(13,15,14,16), (1,5,4,7,2,6,3,8)(9,12)(10,11)(15,16) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 6: $S_3$ 12: $D_{6}$ 24: $S_4$ x 3 48: $S_4\times C_2$ x 3 96: $V_4^2:S_3$ 192: $C_2^3:S_4$ x 2, 12T100 384: 16T747 768: 16T1063 1536: 12T226 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $S_4$
Degree 8: $C_2^3:S_4$
Low degree siblings
16T1535 x 3, 32T205674 x 2, 32T205675 x 2, 32T205676 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1, 2)( 3, 4)(13,14)(15,16)$ |
| $ 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, 2, 2 $ | $24$ | $2$ | $( 1,16)( 2,15)( 3,13)( 4,14)( 5, 9)( 6,10)( 7,12)( 8,11)$ |
| $ 4, 4, 4, 4 $ | $24$ | $4$ | $( 1,15, 2,16)( 3,14, 4,13)( 5, 9, 6,10)( 7,12, 8,11)$ |
| $ 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$ |
| $ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $4$ | $( 5, 8, 6, 7)( 9,12,10,11)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $12$ | $4$ | $( 1, 2)( 3, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,14)(15,16)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $24$ | $4$ | $( 5, 7, 6, 8)( 9,12,10,11)(13,14)(15,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $24$ | $2$ | $( 1,16)( 2,15)( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$ |
| $ 4, 4, 4, 4 $ | $24$ | $4$ | $( 1,15, 2,16)( 3,14, 4,13)( 5,11, 6,12)( 7, 9, 8,10)$ |
| $ 8, 8 $ | $96$ | $8$ | $( 1,10, 3,11, 2, 9, 4,12)( 5,13, 8,16, 6,14, 7,15)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1 $ | $128$ | $3$ | $( 5, 9,15)( 6,10,16)( 7,12,14)( 8,11,13)$ |
| $ 6, 6, 2, 2 $ | $128$ | $6$ | $( 1, 2)( 3, 4)( 5, 9,16, 6,10,15)( 7,12,13, 8,11,14)$ |
| $ 12, 4 $ | $256$ | $12$ | $( 1,13,10, 4,16,12, 2,14, 9, 3,15,11)( 5, 7, 6, 8)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $2$ | $( 9,16)(10,15)(11,14)(12,13)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $48$ | $4$ | $( 1, 2)( 3, 4)( 9,15,10,16)(11,13,12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $24$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,16)(10,15)(11,14)(12,13)$ |
| $ 4, 4, 4, 4 $ | $192$ | $4$ | $( 1,16, 5, 9)( 2,15, 6,10)( 3,13, 7,12)( 4,14, 8,11)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $48$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,14)(10,13)(11,15)(12,16)$ |
| $ 4, 4, 4, 4 $ | $48$ | $4$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,13,10,14)(11,16,12,15)$ |
| $ 8, 4, 1, 1, 1, 1 $ | $96$ | $8$ | $( 5, 8, 6, 7)( 9,16,12,13,10,15,11,14)$ |
| $ 8, 4, 2, 2 $ | $96$ | $8$ | $( 1, 2)( 3, 4)( 5, 8, 6, 7)( 9,15,11,13,10,16,12,14)$ |
| $ 4, 4, 4, 4 $ | $96$ | $4$ | $( 1,16, 7, 9)( 2,15, 8,10)( 3,13, 6,12)( 4,14, 5,11)$ |
| $ 4, 4, 4, 4 $ | $96$ | $4$ | $( 1,15, 8,10)( 2,16, 7, 9)( 3,14, 5,11)( 4,13, 6,12)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 4)( 7, 8)(11,12)(15,16)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $48$ | $4$ | $( 1,16, 2,15)( 3,14, 4,13)( 5, 9)( 6,10)( 7,11)( 8,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $48$ | $2$ | $( 3, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(15,16)$ |
| $ 4, 4, 4, 4 $ | $24$ | $4$ | $( 1,16, 2,15)( 3,14, 4,13)( 5,11, 6,12)( 7,10, 8, 9)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $12$ | $2$ | $( 1,15)( 2,16)( 3,13)( 4,14)( 5,11)( 6,12)( 7,10)( 8, 9)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $12$ | $2$ | $( 1,16)( 2,15)( 3,14)( 4,13)( 5,11)( 6,12)( 7,10)( 8, 9)$ |
| $ 8, 8 $ | $48$ | $8$ | $( 1,10, 3,12, 2, 9, 4,11)( 5,13, 8,15, 6,14, 7,16)$ |
| $ 8, 8 $ | $48$ | $8$ | $( 1,10, 4,11, 2, 9, 3,12)( 5,14, 7,15, 6,13, 8,16)$ |
| $ 6, 3, 3, 2, 1, 1 $ | $256$ | $6$ | $( 3, 4)( 5, 9,15, 6,10,16)( 7,11,14)( 8,12,13)$ |
| $ 6, 6, 2, 2 $ | $256$ | $6$ | $( 1,13,10, 4,15,12)( 2,14, 9, 3,16,11)( 5, 7)( 6, 8)$ |
| $ 4, 4, 2, 2, 1, 1, 1, 1 $ | $48$ | $4$ | $( 3, 4)( 7, 8)( 9,16,10,15)(11,13,12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $48$ | $2$ | $( 1, 2)( 7, 8)( 9,15)(10,16)(11,14)(12,13)$ |
| $ 8, 8 $ | $192$ | $8$ | $( 1,16, 6,10, 2,15, 5, 9)( 3,14, 8,12, 4,13, 7,11)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2 $ | $48$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,14)(10,13)(11,16)(12,15)$ |
| $ 4, 4, 2, 2, 2, 2 $ | $48$ | $4$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,13,10,14)(11,15,12,16)$ |
| $ 8, 2, 2, 2, 1, 1 $ | $192$ | $8$ | $( 3, 4)( 5, 8)( 6, 7)( 9,16,11,13,10,15,12,14)$ |
| $ 4, 4, 4, 4 $ | $96$ | $4$ | $( 1,16, 8, 9)( 2,15, 7,10)( 3,14, 5,11)( 4,13, 6,12)$ |
| $ 4, 4, 4, 4 $ | $96$ | $4$ | $( 1,15, 7,10)( 2,16, 8, 9)( 3,13, 6,12)( 4,14, 5,11)$ |
Group invariants
| Order: | $3072=2^{10} \cdot 3$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |