The Lie group invariance properties of the one‐dimensional radiation hydrodynamic equations with the equilibrium diffusion approximation, a local thermodynamical equilibrium assumption, and an arbitrary material equation of state are derived. These properties are used systematically to generate similarity solutions of these equations for a given form of the equation of state. A comprehensive list of allowed similarity solutions for a perfect gas is presented. Several special cases that have been found previously by other authors appear in the list. Many other cases not reported previously are also presented. An example numerical solution is given for a piston‐driven shock with a thermal precursor.