There is a nonabelian subgroup T of of order 12 generated by elements a, b such that .
What are the generaters a, b for the above group T such that T = <a , b> and what are the elements of T?
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.