Group operation in generator-relator representation Let $G$ be a finite group of order $n$ given by generator-relator representation. **Question :** Let $a$ and $b$ are two elements of $G$ and we want to know the result of $ab$. Is there any way to know the result, without writing down the group as a table? What other information I need to store so that I can get the result of the $ab$($a$ group operation $b$)?

This is called "the word problem", and it is hard. In some finitely presented groups it is actually undecidable. Of course, for finite groups you can always check by hand or write algorithms that terminate, giving you an answer. But it is not much different from "writing down the group as a table" (and in fact it is harder than that).