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