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In this paper we investigate the Fuglede's conjecture for$\mathbb{Z}_{p^2q^2r}$ and provide a proof of it under the condition $p^2q^2\le r$. We develop a new technique by studying the divisibility of the maskpolynomial of a given set by some system of cyclotomic polynomials. Thiscombined with a so-called mod-p-method gives a strong tool to study Fuglede'sconjecture in this and some other cases.

Spectral Tile Direction in the Group $\mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2} \times \mathbb{Z}_r$