The order of an element is the least common multiple of the order of x in S_3 and the order of y in Z_4.

Elements of S_3 (apart from the identity) have order 2 or 3. Elements of Z_4 have order 2 or 4. So to get an element to have order 6, you must choose x to have order 3 and y to have order 2.

Next, if a has order 6 then must have order 2 and so b must have order 4. So if then z must have order 1 or 2 in S_3, and w must have order 4 in Z_4.

That should be enough to get you started.