3-Designs with block size 6 from PSL(2,q)PSL(2,q) and their large sets

We investigate the existence of 3-designs and uniform large sets of 3-designs with block size 6 admitting PSL(2,q)PSL(2,q) as an automorphism group. We show the existence of simple 3-(28,6,10m)(28,6,10m) designs for 1⩽m⩽2301⩽m⩽230. Most of these designs were previously unknown.