Applications of computation on very large groups I have been studying computational group theory and I am reading and trying to implement these algorithms. But what that is actually bothering me is, what is the practical advantage of computing all properties of extremely large groups, moreover it is a hard problem?
It might give birth to new algorithms but does it solve any problem specific to group theory or other branches affected by it?

The existence of several of the large finite simple sporadic groups, such as the Lyons group and the Baby Monster was originally proved using big computer calculations (although I think they all now have computer-free existence proofs). 
Many of the properties of individual simple groups, such as their maximal subgroups and their (modular) character tables, which are essential for a deeper understanding of the groups, have been calculated by computer.
Some significant theorems in group theory have proofs that are partly dependent on computer calculations, usually for small or medium sized special cases that are not covered by the general arguments. A recent example of this is the proof of Ore's conjecture that every element in every finite simple group is a commutator.
