p-elementary group ($E=P\times C,P\text{ a }p\text{-group},C\text{ cyclic},p\nmid |C|$)

A p-elementary group is a direct product of groups of a p-group and a cyclic group whose order is coprime to $p$. Either factor may be trivial. The coprime cyclic factor is central, a stronger requirement than merely having a cyclic subgroup with p-group quotient.