I have a question about the permutations of the symmetry group of the rhombus. The rotational 2 symmetries are easy. I am unsure on the vertical and horizontal lines of symmetry. The corners that lie on the reflection line, are they reflected or do they remain the same - so are the 2 permutations in the image below correct or not ?
The symmetries of a rhombus correspond in a certain sense to the symmetries of a non-square rectangle. Imagine the rectangle portrait. Simply label the top edge $1$, the righthand edge $2$, the bottom $3$, and the lefthand $4$. Then the symmetries are:
The Identity: $$\begin{pmatrix} 1 & 2 & 3 & 4\\ 1 & 2 & 3 & 4\end{pmatrix}.$$
Vertical: $$\begin{pmatrix} 1 & 2 & 3 & 4\\ 1 & 4 & 3 & 2\end{pmatrix}.$$
Horizontal: $$\begin{pmatrix} 1 & 2 & 3 & 4\\ 3 & 2 & 1 & 4\end{pmatrix}.$$
Rotational: $$\begin{pmatrix} 1 & 2 & 3 & 4\\ 3 & 4 & 1 & 2\end{pmatrix}.$$
These are precisely the symmetries you get using your notation for the rhombus. Can you see why?