By computer simulation, we study SU(N) lattice gauge theory with an action containing couplings with the plaquette trace in both the fundamental and adjoint representations of the group. We investigate whether such an action will permit continuation around the phase transition of the Wilson theory when N exceeds four. By extrapolating this transition to where its latent heat vanishes, we locate the corresponding critical points for N up to eight. The phase structure of lattice gauge theory has been of crucial importance to our understanding of quark confinement. The absence of phase transitions in the fourdimensional models based on SU(2) (Creutz 1979) and SU(3) (Creutz and Moriarty 1982b, Ardill et a1 1983a) indicates that the linear interquark potential at large separations persists from the strong-coupling into the weak-coupling domain. Recently, this simple picture acquired a fascinating complication with the discovery of phase transitions in the Wilson theory when the gauge group is SU(N) with N >

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