We studied how one defines the symmetry group of a polyhedron and how to find symmetry groups of the five regular polyhedra(tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron). We show explicitly that orientation-preserving symmetry group of a regular polyhedron can be generated by two elements satisfying some relations and that each element of the symmetry group corresponds to a rotation around some axis.