In this paper, we improve the upper bounds for the genus of the group \(\mathcal{A} = {Z}_{m_1} \times {Z}_{m_2} \times {Z}_{m_3}\) (in canonical form) with at least one even \(m_i\), \(i = 1, 2, 3\). As a special case, our results reproduce the known results in the cases \(m_3 = 3\) or both \(m_2\) and \(m_3\) are equal to \(3\).