Nonabelian groups first appear at order $6$, uniquely as $S_3$.Suggest a slogan
Every group of order at most \(5\) is abelian, and up to isomorphism the only nonabelian group of order \(6\) is the dihedral group \(D_3\cong S_3\).
Groups of order \(1\), \(2\), \(3\), and \(5\) are cyclic, groups of order \(4\) are abelian by FF-TSZNT, and the groups of order \(6\) are classified in FF-TEGRU.
What is the smallest order at which a nonabelian group exists, and which group is it?