It is well known that two elements of are conjugate if and only if they have the same rational canonical form. This leads to a parameterization of the conjugacy classes of by partition valued functions from the set of monic irreducible polynomials over . It is also true that the complex irreducible characters of are parameterized by partition valued functions. In this talk, we will discuss these parameterizations and then describe a method of constructing supercharacter theories of from supercharacter theories of which can be described combinatorially using these parameterizations.