Why Cayley table representation?

Question

I have seen in many times group is given as a input(for algorithms) as a Cayley table form instead of generator and relators. What is the advantage of the Cayley table representation as compare to other representations of the group?

Answer 1

Derek Holt's comment is an answer. Usually it is very hard for a given presentation of a group to conclude certain properties about the group. Theoretically, one can often show that it is undecidable in general, e.g., whether or not the given group is the trivial one, or not. Here are two well-known examples, which can be decided, but which already show that this might be not so convenient.

Question 1 (easier): Is the group with presentation $\langle a,b\mid ab^2a^{-1}b^{-3}=ba^2b^{-1}a^{-3}=e\rangle$ the trivial group or not?

Question 2 (harder): Has the group with presentation $\langle a,b\mid a^{2}=b^{3}=(ab)^{13}=[a,b]^{5}=[a,bab]^{4}=(ababababab^{-1})^{6}=e\rangle$ more than $17000000$ elements, or not? Is it a simple group or not?