  In this paper, we consider the steps to be followed in the analysis and
interpretation of the quantization problem related to the $C_{2,8}$ channel,
where the Fuchsian differential equations, the generators of the Fuchsian
groups, and the tessellations associated with the cases $g=2$ and $g=3$,
related to the hyperbolic case, are determined. In order to obtain these
results, it is necessary to determine the genus $g$ of each surface on which
this channel may be embedded. After that, the procedure is to determine the
algebraic structure (Fuchsian group generators) associated with the fundamental
region of each surface. To achieve this goal, an associated linear second-order
Fuchsian differential equation whose linearly independent solutions provide the
generators of this Fuchsian group is devised. In addition, the tessellations
associated with each analyzed case are identified. These structures are
identified in four situations, divided into two cases $(g=2$ and $g=3)$,
obtaining, therefore, both algebraic and geometric characterizations associated
with quantizing the $C_{2,8}$ channel.
