i am pretty sure there is a theorem to this effect. but note that 6 is the LCM of the orders of all the elements of . it follows that any element raised to it will give the identity.
you can look up the theorem to prove this elegantly, or since is small. just prove it by cases. has elements of orders 1,2, and 3. show that for each of these kinds of elements, the 6th power yields the identity