With every finite group, a permutation polytope can be associated whose vertices are matrices with entries from the set \(\{0, 1\}\). Easy examples show that isomorphic groups may lead to affinely non-equivalent permutation polytopes and there are some non-isomorphic groups whose permutation polytopes are affinely equivalent. So it is interesting to know which groups lead to affinely equivalent permutation polytopes. In the present article, the authors characterize effective equivalence of groups using geometric methods and representation theory (the notion of effective equivalence, which is in fact generalization of group isomorphism, was introduced in an earlier paper by the first author along with others). Here, the authors give a criterion, which determines effective equivalence of two groups in terms of their permutation polyhedrons. Definitions as well as previously proved results are stated and discussed at length that makes the paper self-contained. Many examples are given to make the concepts more understandable and interesting; two unsettled questions are also suggested.