In physics and chemistry, there is the concept of double groups. These are double covers of the usual point groups, obtained by "adding an element $R$, which represents rotation by $2\pi$" to the respective point group.

Since the set $\{ E,R \}$ forms a group isomorphic to $\mathbb Z_2$, it seems that the double group should be some kind of a product of the point group and $\mathbb Z_2$. However, it is not the direct product, since $R$ does not commute with all the elements. Can it be defined a different operation between the point group and $\mathbb Z_2$?