The generalized wreath product of symmetric association schemes was introduced by R.A.Bailey in the European Journal of Combinatorics 27 (2006) 428-435. It is recognized as a unification of both the wreath product and the direct product of symmetric association schemes. While its potential applicability to any association scheme had been implied, this paper provides a formal and explicit confirmation of that claim. Moreover, we establish the irreducible representations of its adjacency algebra and Terwilliger algebra.