In this paper we use facets of mixed-integer sets with two and three variables to derive valid inequalities for integer sets defined by a single equation. These inequalities also define facets of the master cyclic group polyhedron of Gomory. Facets of this polyhedron give strong valid inequalities for general mixed-integer sets, such as the well-known Gomory mixed-integer cut. In particular, our inequalities generalize the slope facets of Araoz, Gomory, Johnson and Evans (2003). In addition, they dominate the strong fractional cuts of Letchford and Lodi (2002).