I know that there are two groups of order 45, and obviously one of them (up to isomorphism) is $\mathbb{Z}_{45}$. I'm trying to understand explicitly what the structure of the other is like.
By Cauchy's Theorem and Sylow's First Theorem, it has a subgroup of order 3, one of order 5, and one of order 9. What this second group of order 45 doesn't have, unlike $\mathbb{Z}_{45}$, is a subgroup of order 15. This rules out an element of order 15, which would generate a cyclic subgroup. I know the group is abelian since we can write it as a product of normal subgroups. Otherwise I'm unsure of what consequences this has, and what the structure of the non-cyclic group of order 45 is.
So basically my question is...how do we obtain an explicit description of the the non-cyclic group of order 45?