Let $G$ be the group of discrete transformations that leave a rectangle invariant (with the composition law given by subsequent application of two transformations as the group product), including the trivial transformation which does not interchange any of its vertices (e), rotations by integer multiples of $\pi$ $(a)$, reflections about the vertical symmetry axis $(b)$, and reflections about the horizontal symmetry axis $c$. Construct the Cayley table of $G$ and identify which group it is.