I'm trying to exhibit two non-isomorphic non-abelian groups of order $243 = 3^5$.
I know that there are subgroups of order $3^k$ for $k$ ranging from $0$ to $5$ inclusive, where the subgroup of order $3$ is cyclic, but unsure how to proceed: I was also taking a look at a similar problem but feel like I might be missing something.
Any hints are appreciated; I was just interested since it's a practice prelim question. Thanks!