On geometric representations of finite groups that have an Abelian subgroup of index 2

In this paper, we study some general properties of geometric representations of finite groups. We prove that the sum of all operators of geometric representations equals to zero for any finite group. Consequently, a one-dimensional trivial representation is not contained in a geometric one of any finite group. Moreover, if a group G has an abelian subgroup A of index two t hen under certain conditions the vectors of its geometric graph from A and G \ A are equidistant from each other. We illustrated our results with examples of generalized quaternion groups, namely Q8 and Q12. We prove in particular that all of maximal dimension complex irreducible representations of these groups may be built from their geometric representations.