The theory of Fuchsian group plays an important role in the study of compact Riemann surfaces Automorphism groups, which was initially studied by A.M. Macbeth. The biholomorphic self transformations of a compact Riemann surface S of genus g (2) forms a finite group whose order cannot exceed 84 (g -1). This maximum bound is attained for infinitely many values of g , the least being 3. The groups for which this bound is attained are called Hurwitz groups, and this groups belong to the class of perfect groups which are non soluble. In context to the class of soluble groups, the corresponding bound is 48 (g -1) and this bound is also attained for infinitely many values of g . The problem of finding such bounds for various sub-classes of the finite soluble groups and the number of values of g for which these bounds are attained has been the theme of many research papers during the last few decades.

In this paper, a set of necessary and sufficient conditions for the existence of smooth epimorphism from a Fuchsian group to the point group O_h, which belongs to the sub-class of octahedral group considering the symmetries of Sulfur-Hexafloride ( S F₆ ) molecule Having the abstract group representation ²= ⁴=() ²=1 is established to fulfill the objective.