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I was doing something which needs to know sizes of all orbits of the stabilizer of two points in a 2-transitive permutation group. Since all 2-transitive permutation groups are known ans so are their stabilizer of two points, it is not to hard to find all the sizes of all orbits of these stabilizers. However, it seems to me that this case by case discuss will not be a short argument, so I would like to ask: are there any known results? In particular, are there formulas for $PSU(3,q).[de]$, $Sz(q).e$ and $Ree(q).e$, where $q=p^f$, $d\mid gcd(3,q+1)$ and $e\mid f$?

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    $\begingroup$ Of course there are known results. For example the sizes of the orbits of the $2$-point stabilizer of ${\rm PSL}(n,q)$ are $1,1,q-1,(q^n-1)/(q-1)-(q+1)$, and it would be possible to write formulae for all of the infinite families (although the affine examples are less straightforward). I think you need say more precisely what you want to know. $\endgroup$
    – Derek Holt
    Commented Apr 2, 2014 at 6:26
  • $\begingroup$ @DerekHolt Thank you! I've edited the question and focus on three almost simple families now. $\endgroup$
    – Binzhou Xia
    Commented Apr 2, 2014 at 22:30

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