"Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$" I have a question that says this:

Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$ according to the fundamental theorem of finitely generated abelian groups.

I would like to see how it is correctly answered. This is not homework; I'd just like to see a proof.

The Smith Normal Form of the matrix $\begin{pmatrix} 5&0&0\\ 0&4&0\\0&0&8\\1&1&1\\\end{pmatrix}$ (which gives a presentation for the group) has diagonal entries 1, 1, 4;
thus the group is isomorphic to $Z_4$.
