Find noncyclic group of order 4 in U(40)
It's a great example for a lot of things. Like if you need a random abelian noncyclic group, look no further. And hello noncyclic p-group. It's also nice that every element has order 2 so you can attach it to other groups and easily visualize what's going on. A lot of problems that arise in the study of abelian group structure can be simplified by working out the case for Z2+Z2 too because a copy of it lives inside every noncyclic abelian group of even order. I stand by my characterization of the Klein 4 as a wonderful creature.