Element prime order (EPO) group

Question

A finite group $G$ is called EPO-group if every non-identity element of $G$ has prime order.

For a given finite group $G$, let $\pi (G)$ denote the set of all prime divisors of $|G|$.

Does there exist non-isomorphic EPO-groups $G_1$, $G_2$ of same order with $|\pi(G_i)|\geq 2$ , $i=1,2$ ?

Answer 1

The smallest examples are the groups denoted (147,4) and (147,5) in the SmallGroups library. They are both semidirect products $Z_7^2:Z_3$.

Answer 2

Yes, finite, non-isomorphic EPO-groups of the same order are possible. See DonAntonio's answer to a question in the recent thread

When will two groups be isomorphic???