A thorough numerical analysis of a recently proposed cubic bond orientational model for Blue Phase III (Longa, L., and Trebin, H.-R., 1993,Phys. Rev. Lett.,71, 2757) is presented. In addition to the standard quadrupolar tensor fieldQαβ(r), describing the cubic space groups of BPI and BPII, a spatially constant fourth-rank hexadecupolar tensorB4αβγδof cubic point group symmetry is used to describe a cubic bond orientational order. While in BPI and BPII both order parameters are present, in BPIII only the hexadecupolar tensor is non-zero. Hence, BPIII is viewed as a phase of long-range cubic order. Within this model distinct phase diagrams are computed up to four stars ofk-vectors, the elements of a star being related by the point group symmetry operations. In particular, it is possible to account for some of the details found experimentally, such as the dominance of BPI over BPII for high chiralities. If, however, the artificial body centred cubic structure O5is being made metastable, then BPI also vanishes from the phase diagram.