Primitive one-factorizations and the geometry of mixed translations

We construct an infinite family of one-factorizations of KvKv admitting an automorphism group acting primitively on the set of vertices but no such group acting doubly transitively. We also give examples of one-factorizations which are live  , in the sense that every one-factor induces an automorphism, but do not coincide with the affine line parallelism of AG(n,2)AG(n,2). To this purpose we develop the notion of a “mixed translation” in AG(n,2)AG(n,2).