A construction of designs on n+1 points from multiply homogeneous permutation groups of degree n

A simple criterion to construct a t-design on n+1 points from a t-ply homogeneous permutation group of degree n using some orbits of the group is obtained. The design is not simple in general. Applying the criterion to 2-dimensional projective special linear groups PSL(2,q) for q⩾19 acting on the projective line of q+1 points, simple 3- designs are obtained if q≡3 or 7 (mod 12), and simple 3- designs are obtained if q≡3 (mod 4).