Semi-magic dihedral squares

Let $Γ$ be a group of order n² and SMS_Γ(n)=(ai,j)n× n be an n× n array whose entries are all distinct elements of $Γ$. If there exists an element μ∈Γ such that for every row i, there exists an ordering of elements such that ai,j₁ ai,j₂ a_i,jₙ₋₁ ai,jₙ = μ and for every column j there exists an ordering of elements such that ai₁,j ai₂,j a_iₘ₋₁,j aiₘ,j = μ, then SMS_Γ(n) is called a $Γ$-semi-magic square of side n and $μ$ is called a magic constant. We provide a complete characterization of semi-magic squares of side n whose entries belong to a dihedral group Dₖ. Moreover, we show that in our constructions a single semi-magic square may admit two distinct magic constants, depending on the order in which the products are computed.