PGL(2,9) is not isomorphic to S6
This non-isomorphism is important to keep in mind because PSL(2,9) is isomorphic to A6, and thus both these groups have a subgroup of index two that is isomorphic (namely, A6 in S6 and PSL(2,9) in PGL(2,9)). There is also a third group, Mathieu group:M10, that has a subgroup of index two isomorphic to alternating group:A6 that is not a direct factor. For more on this perspective, see supergroups of alternating group:A6.
Proof using conjugacy class sizes
Using the element structure of projective general linear group:PGL(2,9), we see that the conjugacy class size statistics are:
The conjugacy class sizes do not match up, so the groups are not isomorphic.
To show non-isomorphism, it is not necessary to compute the full conjugacy class size statistics of both groups, but we need to compute enough to show that the conjugacy class size statistics cannot match up.