A family of groups P^(m,n) parametrised by non-negative integers m>n is studied generalising the Poincare group P^(1,1). The action of finite-dimensional irreducible real representations of SL(2, C) is used to form semi-direct products. The authors define the complete list of unitary irreducible representations for each P^(m,n) by finding all sub-groups of SL(2, C) which are little groups; some of the subgroups do not occur in the Poincare group. The geometry of the SL(2, C) action and the classification of SL(2, C) invariant tensors is considered. These groups are appropriate symmetry groups for field theories in higher dimensions and generalise the notion of elementary relativistic quantum systems.