A class of groups A(n) having two generators and two relations is studied. This class is of interest for two reasons. It not only involves Fa'b'c groups, and, also for the conjucture given in the study of Fa'b'c groups, but also adds examples to the known class of finite groups with two-generator two-relation presentations. In this paper the order of the groups A(n) are determined and the Reidemeister-Schreier algorithm is used to show that the groups are finite and not metabelian.