If $G$ is a non-cyclic group of order $n^2$, then $G$ is isomorphic to $\mathbb{Z_n} \oplus \mathbb{Z_n}$ I've independently come up with a question (I know it's been asked before, but I can't find the question online) involving the external direct product, non-cyclic groups and isomorphisms.  So, is the following statement true?  
Claim:  "If $G$ is a non-cyclic group of order $n^2$, then $G$ is isomorphic to $\mathbb{Z_n} \oplus \mathbb{Z_n}$."
Where $\mathbb{Z_n} \oplus \mathbb{Z_n}$ is the external direct product of $\mathbb{Z_n}$ and $\mathbb{Z_n}$.
I've been thinking about this for a few hours, but really can't figure out a good place to start (or a counter-example).  

Asking what every non-cyclic group of a given order looks like is just as hard as asking what every group of a given order looks like. This is a very hard problem. For example see this article.
Now if you wanted to know what every abelian group of a given order looks like that is possible. 
