We know that if $n$ is an odd natural number then the dihedral group $D_{2n}$ is isomorphic to the group $D_n\times \mathbb{Z}_2$ where $D_m$ is the dihedral group of order $2m$, which has presentation $$ D_m:=\langle r, f\rangle =\langle r, f : r^m = f^2 = (rf)^2 =e \rangle \, . $$
My question is: what is the relation between $D_{2n}$ and $D_n$ if $n$ is even? Is there any such thing?
Thank you.