Resolvable group divisible designs with block size four and index three

In this paper, we investigate the existence of resolvable group divisible designs (RGDDs) with block size four, group-type hnhn and index three. The necessary conditions for the existence of such a design are n⩾4n⩾4 and hn≡0hn≡0(mod4). These necessary conditions are shown to be sufficient except for (h,n)∈{(2,4),(2,6)}(h,n)∈{(2,4),(2,6)} and possibly excepting (h,n)=(2,54)(h,n)=(2,54).