In chemistry, point groups are used to describe the molecular symmetries of a molecule. Meanwhile, a crystallographic point group is a set of point group that holds at least one point in a fixed position and have some restriction on the rotational symmetries. In this research, the application of graph theory and group theory on the symmetry study of molecules is presented in which the Cayley graph of crystallographic point groups under the hexagonal crystal system is determined. The generators are also computed for groups