  Properties of a given symmetry group G are very important in investigation of
a physical system invariant under its action. In the case of finite spin
systems (magnetic rings, some planar macromolecules) the symmetry group is
isomorphic with the dihedral group D_N. In this paper group-theoretical
`parameters' of such groups are determined, especially decompositions of
transitive representations into irreducible ones and double cosets. These
results are necessary to construct matrix elements of any operator commuting
with G in an efficient way. The approach proposed can be usefull in many
branches of physics, but here it is applied to finite spin systems, which serve
as models for mesoscopic magnets.
