In this talk, we apply the complex of groups framework developed by Bridson and Corsen to compute the homology of a simplicial complex acted upon regularly by a finite cyclic group. This framework compresses the original complex into its quotient simplicial complex and additional local data. By combining this local data with suitable boundary matrices of the quotient, we have constructed a matrix over a group ring for each dimension. We call them -boundary matrices and we will show that the homology of the original simplicial complex can be obtained by decomposing the -boundary matrices. This algebraic algorithm complements the geometric compression algorithm by Carbone, Nanda, and Naqvi.