As far as I know CGT is mostly of interest to mathematicians, because there are some groups whose study require calculations that are absolutely impractical to carry out by hand, such as presentations of the sporadic groups. Or maybe you need to know how many nonabelian groups of order 2652 are there and throwing it to a computer is just the fastest way to find out or you need to compute conjugacy classes or stuff like that.

Computational algebra more broadly however has many applications to cryptography for example