Classification of triply transitive finite groups

A permutation group $G$ on a set $X$ is said to be $k$-transitive if it is both transitive on $X$ and either $k=1$ or the point stabilizer $G_x$ is $(k-1)$-transitive on $Xsetminus{x}$.

Is there a classification of 3-transitive finite groups?

Examples:

• For every non-negative integer $n$, $S_n$ and $A_n$ are 3-transitive on ${1,2,dots,n}$
• If $n-1=p^f$ is a prime power, then all $G$ with $operatorname{PGL}(2,p^f) leq G leq operatorname{PGamma L}(2,p^f)$ are 3-transitive
• If $n-1=q^2$ is the square of a prime power, then also all $G$ with $M(q^2)leq G leq operatorname{PGamma L}(2,p^f)$, where $operatorname{PSL}(2,q^2) < M(q^2) < operatorname{PGamma L}(2,q^2)$

Every sharply triply transitive group is either $operatorname{PGL}(2,p^f)$ or $M(q^2)$, of order $((n-1)^2-1)((n-1)^2-(n-1))/(n-2) = n(n-1)(n-2)$. This is due to Zassenhaus; see Huppert–Blackburn (XI.1.4.b, XI.2.1, and XI.2.6). However, there are triply transitive groups that are not sharply triply transitive (such as $operatorname{PGamma L}(2,p^f)$ for $f>1$).

If $n$ is odd, then Wagner (1966) showed that any non-identity normal subgroup of a triply transitive group is also triply transitive. By taking a minimal normal subgroup (and then a minimal normal subgroup of that) we get a simple triply transitive group of the same degree, so if we are only interested in $n$, then we need only consult our knowledge of finite simple groups.

I think $operatorname{ASL}(n,2)$ is always triply transitive.

Here are the triply transitive groups of degree $n < 2500$ that don’t fall into the above categories:

• $M_{11}$ of degree 11
• $M_{11}$ of degree 12, $M_{12}$ of degree 12
• $A_7 ltimes 2^4$ of degree 16 (?)
• $M_{22}, operatorname{Aut}(M_{22})$ of degree 22
• $M_{23}$ of degree 23
• $M_{24}$ of degree 24

Note these are mostly Mathieu groups.

Bibliography

• Wagner, A.
  “Normal subgroups of triply-transitive permutation groups of odd degree.”
  Math. Z. 94 (1966) 219–222
  MR199251
  DOI:10.1007/BF01111350
• Siemons, Johannes.
  “Normal subgroups of triply transitive permutation groups of degree divisible by 3.”
  Math. Z. 174 (1980), no. 2, 95–103.
  MR592907
  DOI:10.1007/BF01293530