In a distributed space-time coding scheme, based on the relay channel model, the relay nodes co-operate to linearly process the transmitted signal from the source and forward them to the destination such that the signal at the destination appears as a space time block code. Recently, a code design criteria for achieving full diversity in a partially-coherent environment have been proposed along with codes based on differential encoding and decoding techniques. For such a set up, in this paper, a nondifferential encoding technique and construction of distributed space time block codes from unitary matrix groups at the source and a set of diagonal unitary matrices for the relays are proposed. It is shown that, the performance of our scheme is independent of the choice of unitary matrices at the relays. When the group is cyclic, a necessary and sufficient condition on the generator of the cyclic group to achieve full diversity and to minimize the pairwise error probability is proved. Various choices on the generator of cyclic group to reduce the ML decoding complexity at the destination is presented. It is also shown that, at the source, if non-cyclic abelian unitary matrix groups are used, then full-diversity can not be obtained. The presented scheme is also robust to failure of any subset of relay nodes.