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The group $\PGL(2,q)$, $q=p^n$, $p$ an odd prime, is
$3$-transitive on the projective line and therefore it can be
used to construct $3$-designs. In this paper, we determine the
sizes of orbits from the action of $\PGL(2,q)$ on the $k$-subsets
of the projective line when $k$ is not congruent to $0$ and 1
modulo $p$. Consequently, we find all values of $\lambda$ for
which there exist $3$-$(q+1,k,\lambda)$ designs admitting
$\PGL(2,q)$ as automorphism group. In the case $p\equiv 3
\pmod{4}$,
the results
and some previously known facts are used
to classify 3-designs from $\PSL(2,p)$ up to isomorphism.
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