Multiplication in factor groups is easy. It is defined as
$\displaystyle (aH)(bH) = abH$.
For example Let $\displaystyle G=\mathbb{Z}, H = Z\mathbb{Z}$. Since G is abelian, H is normal in G.
Consider two cosets of $\displaystyle 4\mathbb{Z}$, $\displaystyle (1+ 4\mathbb{Z}) +(2+4\mathbb{Z}) = (3 + 4\mathbb{Z})$.
Now in your example, we take $\displaystyle [(12)H][(24)H] = (14)H$. since $\displaystyle (12)(24) = (14)$