$(\mathbb Z_4\times\mathbb Z_6)/\langle(2,3)\rangle$ is isomorphic to $\mathbb Z_2\times\mathbb Z_6$ or $\mathbb Z_{12}$ ?
I think $(\mathbb Z_4\times\mathbb Z_6)/\langle(2,3)\rangle$ is not cyclic group but $\mathbb Z_{12}$ is cyclic.
I cannot construct a homomorphism $\phi:\mathbb Z_4\times\mathbb Z_6\to\mathbb Z_2\times\mathbb Z_6$ such that $\ker\phi=\langle(2,3)\rangle$.