Phase transitions for liquid crystals are typically characterized in terms of bifurcations of critical points for a free energy function. In a widely used model for the free energy due to Katriel et al. (1986) natural symmetries arise from the action of the group  SO(3) wr Z_2 (wreath product) acting on  5x5  matrices through left/right multiplication and transposition.  We investigate invariants for this group action, and also describe some bifurcation behaviour after a standard reduction is made to more tractable  2x2 matrices and the finite group  D_3 wr Z_2 .