  After reviewing the group structure and representation theory for the
dihedral group $D_{2n},$ we consider an intertwining operator $\Phi_\rho$ from
the group algebra $\mathbb{C}[D_{2n}]$ into a corresponding space of semi-magic
matrices. From this intertwining operator, one obtains the generating function
for enumerating the associated semi-magic squares with fixed line sum and an
algebra extending the circulant matrices. While this work complements the
approach to $D_{2n}$ through permutation polytopes, we use only methods from
representation theory.
