Non abelian groups with a prime number of one-dimensional representations

Question

I am interested in examples of nonabelian groups that are "finitely generated by elements of finite order" that have a prime number of one dimensional representations.

For $p=2$ we have $S_N$ for $N\geq 2$. For $p=3$ we have $A_4$.

What about other primes?

I am just as interested in small $p$ examples and general results. I am not looking for complete classifications for all $p$ or indeed any $p$.

I understand that there is an "at least $p$" result for groups of order $p^n$ but I am looking for equality rather than inequality.

Answer 1

As Derek Holt says we can take the direct product of a cyclic group and a perfect group.