In this paper, we provide a classification of the conjugacy classes in the special affine group Qd(p) = (Z_p x Z_p) x SL(2,p) for odd primes. We use the known conjugacy class structure in SL(2,p) to lift elements to their conjugacy class in Qd(p). More specifically, for an element in Qd(p), its conjugacy class depends on whether the matrix component of the element is the identity, has eigenvalue 1, which is split into two classes in SL(2,p), or whether it lies in some other conjugacy class in SL(2,p). In addition, we provide formulas, in terms of p, for both the number and sizes of conjugacy classes in Qd(p).

The authors are sincerely grateful for the Simons Foundation Scheme for the Provision of Magma at US Educational and Scientific Research Organizations, which provided access to the Magma Computer Algebra system.