Hello, I was wondering if I could get some help with this question.

Let G be the subgroup of $\displaystyle S_4 $ generated by (1 2),(3 4) and (1 3)(2 4). Let H be the subgroup of S_4 generated by (1 2) and (3 4). Show that H is a normal subgroup of G. Show that H has order 4 and G/H has order 2. Deduce that G has order 8.

I was wondering if there is a quick way to show that G has order 4 without doing it explicitly? I was also wondering how to show that G/H has order 4. I guess that Lagrange's Theorem proves the last part. Any help with this would be appreciated.