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Monte Carlo sampling of the canonical distribution presents a formidable challenge when the potential energy landscape is characterized by a large number of local minima separated by high barriers. The principal observation of this work is that the multiple local minima and energy barriers in a landscape can often occur as a result of discrete symmetries in the potential energy function. A new Monte Carlo method is proposed, group action Markov chain Monte Carlo (GA-MCMC), which augments more conventional trial moves (e.g. random jumps, hybrid Monte Carlo, etc.) with the application of a group action from a well-chosen generating set of the discrete symmetry group; the result is a framework for symmetry-adapted MCMC. It is shown that conventional trial moves are generally optimal for "local mixing" rates, i.e. sampling a single energy well; whereas the group action portion of the GA-MCMC trial move allows the Markov chain to propagate between energy wells and can vastly improve the rate of "global mixing". The proposed method is compared with standard jumps and umbrella sampling (a popular alternative for energy landscapes with barriers) for potential energies with translational, reflection, and rotational symmetries. GA-MCMC is shown to consistently outperform the considered alternatives, even when the symmetry of the potential energy function is broken. The work culminates by extending GA-MCMC to a clustering-type algorithm for interacting dielectric polymer chains. Not only does GA-MCMC again outperform the considered alternatives, but it is the only method which consistently converges for all of the cases considered. Some new and interesting phenomena regarding the electro-elasticity of dielectric polymer chains, unveiled via GA-MCMC, is briefly discussed.