  Much has been written on reciprocity laws in number theory and their
connections with group representations. In this paper we explore more on these
connections. We prove a "reciprocity Law" for certain specific representations
of semidirect products of two cyclic groups which is in complete analogy with
classical reciprocity laws in number theory. In fact, we show that the
celebrated quadratic reciprocity law is a direct consequence of our main
theorem applied to a specific group. As another consequence of our main theorem
we also recover a classical theorem of Sylvester. Our main focus is on explicit
constructions of representations over sufficiently small fields. These
investigations give further evidence that there is still much unexplored
territory in connections between number theory and group representations, even
at an elementary level.
