1-Designs from the group $PSL_{2} (59)$ and their automorphism groups

In this paper, we consider the projective special linear group $PSL_2 (59)$ and construct some 1-designs by applying the Key-Moori method on $PSL_2 (59)$. Moreover, we obtain parameters of these designs and their automorphism groups. It is shown that $PSL_2 (59) $ and $PSL_2 (59):2$ appear as the automorphism group of the constructed designs.