Explore the concept of finite commutative hypergroups in this comprehensive mathematics lecture. Learn about the precise definition and five key examples: the class hypergroup of a finite group, the character hypergroup of a finite group, the hypergroup associated with a distance-transitive graph, double coset hypergroups arising from Gelfand pairs, and automorphism orbit hypergroups. Gain insights into how these important families can be applied as potential tools for physicists and chemists in Diffusion Symmetry. Delve into early hypergroup history, examine 2D representations of S3 over finite fields, and connect abstract concepts to computational reality through concrete calculations and examples.