Procedures based on group representation theory, allowing the exploitation of geometrical symmetry in symmetric Galerkin BEM formulations, are developed for 3D Neumann elastostatic problems, considered as model problems. Both Abelian and non-Abelian finite symmetry groups are considered. More- over, the formulation is developed under the weaker assumption of partial geometrical symmetry, where the boundary has two disconnected components, one of which is symmetric. Numerical examples for partial geometrical symmetry and Abelian and non-Abelian groups demonstrate the effectiveness of the approach.