Multicomponent spin models with transitive symmetry groups

A number of new results concerning the multicomponent spin systems studied in the first two papers of this series is derived. These results allow for the determination of all such models with number of components M ⩽ 10. There are shown to be 41 such models, each of which is given in terms of its (permissible) symmetry group. The subgroup structure, isomorphisms and the existence of associated algebras of pair energy functions are all discussed in detail.