Using noncommutative geometry we do U(1) gauge theory on the permutation group S₃. Unlike usual lattice gauge theories the use of a non–Abelian group here as spacetime corresponds to a background Riemannian curvature. In this background we solve spin 0, 1 2 and spin 1 equations of motion, including the spin 1 or ‘photon’ case in the presence of sources, i.e. a theory of classical electromagnetism. Moreover, we solve the U(1) Yang–Mills theory (this differs from the U(1) Maxwell theory in noncommutative geometry), including the moduli space of flat connections. We show that the Yang–Mills action has a simple form in terms of Wilson loops in the permutation group, and we discuss aspects of the quantum theory.