Show that the group of permutations of $\{1,2,3,4\}$ $$\sigma_4$$ is equal to the product of it's subgroups $$C_2\times C_2 $$ and$$D_6=(x^3=y^2=1, yx=x^2y)$$
I'm not sure whether to just multiply the groups and find all $24$ terms or whether there is a short cut using the semi-direct product I know $$Aut(C_2\times C_2)\cong D_6 $$