The only two remaining possibilities then are that "ab" = "d" or that "ab" = "f"; we would expect these two guesses to each have the same outcome, up to isomorphism, because "d" and "f" are inverses of each other and the letters that represent them are inherently arbitrary anyway. So without loss of generality, take "ab" = "d". If we arrive at another contradiction, we must assume that no group of order 6 has the identity skeleton we started with, as we will have exhausted all possibilities.