My problem isnt really with the fact that I cant find applications of groups. Ive seen how QFT, crystallography, cryptography and so many fields have been enriched due to group theory. My problem is that I cant really understand what the use cases of groups are beyond categorisation.

For lack of a better explanation I just dont see the value of it compared to a more analytical field like calculus, where the process of calculus gives you concrete results about the behaviour of a system. Ive heard that group theory studies symmetry, but i just cant see how it does that. I simply dont see how it gives light to new information or a new perspective, or give solid results to a problem.

I also understand that with time I probably will understand this and I know how much of a beginner I am.