According to my lecture book (Linear Representations of Finite Groups by Serre), the dihedral group $D_n$ consists of $n$ rotations and $n$ reflections in the plane that preserve a regular polygon with $n$ vertices. He later writes that if $D_n$ is realized as a group of rotations and reflections of $3$-space then $D_{nh}$ can be realized as the group generated by $D_n$ and the reflection through the origin. Edit: The book mentions that the order of $D_{nh}$ is $4n$ and it is the product $D_n\times I$ where $I$ is the group $\{1,\iota\}$ with $\iota^2=1$.
First of all, I would like to know what he means by "3-space", since a polygon is a 2-dimensional object.
Second, I would like to be sure that I understand the groups correctly. I understand $D_n$ to have the $n$ reflections where the $n$ symmetry elements are lines in the plane, so that for example the 4 symmetry elements of reflections of a regular square are the ones depicted on the figure below on the left. I then assume that $D_{nh}$ is what we get if we add a reflection with the origin as the symmetry element, as depicted on the right. Is that correct?
Edit: I know many here hate screenshots, but from the comments I think it makes sense in this case.
