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In this paper, we define a set which has a finite group action and isgenerated by a finite color set, a set which has a finite group action, and asubset of the set of non negative integers. we state its properties to applyone of solution of the following two problems, respectively. First, wecalculate the generating function of the character of symmetric powers ofpermutation representation associated with a set which has a finite groupaction. Second, we calculate the number of primitive colorings on some objectsof polyhedrons. It is a generalization of the calculation of the number ofprimitive necklaces by N.Metropolis and G-C.Rota.

Symmetric powers of permutation representations of finite groups and primitive colorings on polyhedrons