Let $G = C_m \square P_n$ (grid with $m$ rows, $n$ columns, where bottom and top row are connected via edges). What are all of the possible symmetries of $G$? Equivalently, I would like to describe the automorphism group of $G$. I can rotate about the cycle, flip about the horizontal, flip about the vertical but I can also do many combinations of these elementary actions.
I ultimately want to determine all possible group actions so that I can use Burnside's Lemma/Polya's Theory to determine the number of nonequivalent colouring of $G$ with, lets say, 3 colours. Once I have the group, the latter will be straightforward.
We can distinguish the first and last column from the others by their degree: vertices in the first and last column have degree $3$, all the rest have degree $4$. So if we label the vertices $v_{ij}$ where $1 \le i \le m$, $1 \le j \le n$, then an automorphism sends $v_{11}$ either to $v_{i1}$ for some $i$ or to $v_{in}$ for some $i$. Furthermore, $v_{21}$ can be sent to $v_{i\pm1,1}$ (or respectively $v_{i\pm1,n}$) since it must be sent to a degree-$3$ vertex adjacent to the image of $v_{11}$.
Once $f(v_{11})$ and $f(v_{21})$ are determined, that determines the rest of the automorphism. (First we have a unique way to place $v_{31}, v_{41}, \dots, v_{m1}$; then once the first column is placed, that determines the second column's vertices by their adjacencies to the first; then we go through the columns one by one until they've done.)
So find an automorphism for every possibility of $f(v_{11})$ and $f(v_{21})$, and you've found all the automorphisms.