Does there exists $x_1$, $x_2$ , $y_1$ , $y_2 \in \mathbb{Z}^2$, such that
$$\frac{\mathbb{Z}^2 \oplus \mathbb{Z} ^2 \oplus \mathbb{Z} ^2 }{ \langle (x_1,y_1,0), (0,x_1,y_1) , (x_2,y_2,0), (0,x_2,y_2) \rangle} \cong \mathbb{Z}^2 \oplus \mathbb{Z_m}$$ for some $m > 1$?
So far, I can find quotients of the form $\mathbb{Z}^2$ and $\mathbb{Z}^2 \oplus \mathbb{Z_{m_1}} \oplus \mathbb{Z_{m_2}}$ where $m_1$ and $m_2$ are not coprime.
Any hint or idea will be very appreciated, thank you!