Consider a cartesian \((x y z)\) coordinate system and define the following operations: \(R=\) rotation by \(\pi\) in the \(x-y\) plane, \(E=\) do nothing, \(I=\) inversion of all three axes, and \(\sigma=\) reflection in the \(x-y\) plane. Show that these operations form a group under the product of transformations. Show that this group contains subgroups \(S_{2}=\{E, I\}\) (inversion subgroup) and \(C_{2}=\{E, R\}\) (cyclic subgroup), and that the full group can be written as a direct product of these subgroups.