Steiner systems $S(5,6,v)$ with $v=72$ and $84$

It is proved that there are precisely 4204 pairwise non-isomorphic Steiner systems $S(5,6,72)$ invariant under the group $\mathrm{PSL}_2(71)$ and which can be constructed using only short orbits.

It is further proved that there are precisely 38717 pairwise non-isomorphic Steiner systems $S(5,6,84)$ invariant under the group $\mathrm{PSL}_2(83)$ and which can be constructed using only short orbits.