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I don't know exactly what information you need, but if that includes the actual calculation of the homology:

Look at Hilton and Stammbach's A COurse Course in Homological Algebra. In Theorem 15.2 they prove Künneth's theorm for homology of groups with coefficients in $\mathbb Z$ which reduces the computation of the homology of a direct product of groups to the computation of the homology of the factors and certain $\mathrm{Tor}$s between them. Using the structure theorem for finitely generated abelian groups, the computation of homology for cyclic groups (Proposition VI.7.1 for the finite ones, section VI.4 and corollary VI.5.6 for the infinite ones), and straightforward computations of $\mathrm{Tor}$s between cyclic abelian groups, you are done.

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I don't know exactly what information you need, but if that includes the actual calculation of the homology:

Look at Hilton and Stammbach's Introduction to A COurse in Homological Algebra. In Theorem 15.2 they prove Künneth's theorm for homology of groups with coefficients in $\mathbb Z$ which reduces the computation of the homology of a direct product of groups to the computation of the homology of the factors and certain $\mathrm{Tor}$s between them. Using the structure theorem for finitely generated abelian groups, the computation of homology for cyclic groups (Proposition VI.7.1 for the finite ones, section VI.4 and corollary VI.5.6 for the infinite ones), and straightforward computations of $\mathrm{Tor}$s between cyclic abelian groups, you are done.

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I don't know exactly what information you need, but if that includes the actual calculation of the homology:

Look at Hilton and Stammbach's Introduction to Homological Algebra. In Theorem 15.2 they prove Künneth's theorm for homology of groups with coefficients in $\mathbb Z$ which reduces the computation of the homology of a direct product of groups to the computation of the homology of the factors and certain $\mathrm{Tor}$s between them. Using the structure theorem for finitely generated abelian groups, the computation of homology for cyclic groups (Proposition VI.7.1 for the finite ones, section VI.4 and corollary VI.5.6 for the infinite ones), and straightforward computations of $\mathrm{Tor}$s between cyclic abelian groups, you are done.