First, S3 has order 6 while has order 2 (g just swaps 2 and 3 so doing g twice swaps back and gives the identity) so that there are 6/2= 3 left cosets. Further, those cosets partition S3 so there are, in fact three left cosets. One, the one containing the identity, is just H itself. is not in H and so generates another coset: taking it with the identity gives itself and taking it with gives . Can you find the third coset?