1-Designs Constructed from the Groups $PSL_2(81)$ and $PSL_2(89)$

Abstract

In this paper, some designs from the primitive permutation representations of the groups $PSL_2(81)$ and $PSL_2(89)$ are constructed using the Key-Moori Method 1. We determine the automorphism groups of all the obtained designs and prove that the groups $PSL_2(81)$, $PSL_2(81).2$, $PSL_2(81):2$, $PGL_2(81)$,  $PΣL_2(81)$, $PΓL_2(81)$, $PSL_2(89)$ and $PSL_2(89):2$ appear as the automorphism groups of these constructed designs.