There are cryptographic protocols being developed for non-abelian groups. For some protocols it is necessary to know whether the group has an efficient representation as a matrix group (say, a matrix group over a field $\mathbb{F}$).
What should I do to find out whether a semidirect product of finite groups can be represented as a matrix group efficiently?
Particularly, say semidirect products of the form $(\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes_{\phi} \mathbb{Z}_q$, where $p,q$ are distinct primes, $(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes_{\theta} \mathbb{Z}_3$. How can I check whether they have efficient, faithful representations as matrix groups (provided that I know $\phi,\theta$)?
Thanks a lot in advance.