Groups of order 15

Let \(\abs G=15\). By Sylow's theorems, the number of Sylow \(3\)-subgroups divides \(5\) and is congruent to \(1\) modulo \(3\), and the number of Sylow \(5\)-subgroups divides \(3\) and is congruent to \(1\) modulo \(5\); both numbers are \(1\). So \(G\) has normal subgroups \(P\cong\ZZ/3\ZZ\) and \(Q\cong\ZZ/5\ZZ\) with \(P\cap Q=\theset{e}\) and \(\abs{PQ}=15\), and FT-7NMQR gives \(G\cong\ZZ/3\ZZ\times\ZZ/5\ZZ\cong\ZZ/15\ZZ\).