I've used computers to do some group theory calculations. I was too lazy to figure out how to do them in GAP, so I wrote the programs I needed in C++. One particular paper that I used them in is my paper "The Picard Group of the Moduli Space of Curves with Level Structures". The paper and code are available on my webpage at http://www-math.mit.edu/~andyp/papers/PicardGroupLevel.html

In that paper, I needed to know some twisted group cohomology groups. The groups depended on a parameter $g$, but I could prove that they were independent of the parameter once $g$ was large, so I was able to reduce myself to computing a finite number of cases on a computer.