Summary: Let \(G\) be a finite group and let \(k \geqslant 2\) be an integer. A \((G, k, 1)\)-difference matrix (DM) is a \(k \times |G|\) matrix \(D = (d_{ij})\) with entries from \(G\), such that for all distinct rows \(x\) and \(y\), the multiset of differences \(\{d_{xj}d_{yj}^{-1} : 1 \leqslant j \leqslant|G|\}\) contains each element of \(G\) exactly once. Let \(H\) be a finite abelian group and let \(D_{2H} = \langle H, b \mid b^2 = 1, bhb = h^{-1}, h\in H\rangle\) be the generalized dihedral group of \(H\). It is proved that a \((D_{2H}, 4, 1)\)-DM exists if and only if \(H\) is of even order and \(H\) is not isomorphic to \(\mathbb{Z}_4\). Also for the nine non-abelian groups \(G\) of order 16, we obtain \((G, 6, 1)\)-DMs over five of them, \((G, 5, 1)\)-DMs over three of them and a \((G, 4, 1)\)-DM over one of them. No \((G, 5, 1)\)-DM exists for this last group, where \(G = Q_{16}\).