3-Designs from PSL(2,q)PSL(2,q)

The group PSL(2,q)PSL(2,q) is 3-homogeneous on the projective line when q   is a prime power congruent to 3 modulo 4 and therefore it can be used to construct 3-designs. In this paper, we determine all 3-designs admitting PSL(2,q)PSL(2,q) with block size not congruent to 0 and 1 modulo p   where q=pnq=pn.