In the notion of nilpotent group, a finite group $G$ is nilpotent $\iff$ every Sylow subgroup is normal in $G$ $\iff$ $G$ is isomorphic to direct products of all Sylow subgroups of $G$.
Under what conditions is $G$ cyclic?
I have trouble in the case that $G\cong P_{1}\times P_{2}\times P_{3}$, where each $P_{i}$ are the Sylow subgroups of $G$ of order $4,27,5$, respectively.