  In this paper we prove some general theorems about representations of finite
groups arising from the inner semidirect product of groups. We show how these
results can be used for standard applications of group theory in quantum
chemistry through the orthogonality relations for the characters of irreducible
representations. In this context, conditions for transitions between energy
levels, projection operators and basis functions were determined. This approach
applies to composite systems and it is illustrated by the dihedral group
related to glycolate oxidase enzyme.
