Simple expressions of $\frac{SU(p) \times SU(p)}{\mathbb{Z}_p}$?

Question

We know that properties of $$\frac{SU(2) \times SU(2)}{\mathbb{Z}_2}={SO(4)}$$

1. Is there a nice group isomorphism of $$ \frac{SU(p) \times SU(p)}{\mathbb{Z}_p}? $$ for $p=3,4$?

2. If we include additional $U(1)$, do we have a nicer isomorphism to a familiar group?
   $$ \frac{SU(p) \times SU(p) \times U(1)}{(\mathbb{Z}_p)^2}? $$ for $p=2,3,4$?

All the mod factors are the centralizer subgroups.

Thank you <3.