Orbits of
![$PSL(2,p)$](img1.png)
on 6-element subsets of the projective line with
prescribed non-trivial stabilizer are described. A refinement of cross-ratio
computations to
![$PSL(2,p)$](img1.png)
orbits allows to determine the orbits on
5-element subsets that they cover. Then Steiner
![$5$](img2.png)
-
![$(p+1,6,1)$](img3.png)
designs
are assembled from them. In particular, there is one isomorphism type
of
![$5$](img2.png)
-
![$(48,6,1)$](img4.png)
designs that consists of
![$PSL(2,p)$](img1.png)
orbits of the same size,
each being a
![$3$](img5.png)
-
![$(48,6,30)$](img6.png)
design. There are 7 isomorphism types
of
![$5$](img2.png)
-
![$(84,6,1)$](img7.png)
designs of this type. Generally, Steiner
![$5$](img2.png)
-
![$(p+1,6,1)$](img3.png)
designs
with such an orbit partition may only exist if
![$p\equiv 48,\ 84\ mod \ 180$](img8.png)
.