The connection between the order of simple groups and the maximum number of elementary particles

The aim of this article is to present spherical, Euclidean and hyperbolic polyhedra and find some connections of the order of their reflection groups and simple groups such as PGL(2, 7), PGL(2, 8), PGL(2, 7) × C2, PSL(2, 31) × C2 to the number of elementary particles. In the present work we show that a larger number of 72 or 84 elementary particles is consistent with super string theory, M-theory and heterotic string theory. The philosophy of the work is based on El Naschie's E-infinity interpretation of Emmy Nöther's theorem.