Example of primitive permutation group with a regular suborbit and a non-faithful suborbit

Question

I would like some examples of groups $G$ satisfying all of the following criteria:

1. $G<Sym(n)$, the symmetric group on $n$ letters, and $G$ is primitive.
2. $G$ has a regular suborbit, i.e. if $M$ is a point-stabilizer, then there is an orbit of $M$ on which $M$ acts regularly.
3. $G$ has a non-faithful suborbit, i.e. if $M$ is a point-stabilizer, then there is an orbit of $M$ on which $M$ acts non-faithfully.

Just to be clear: the suborbits in 2 and 3 need to be different! If possible, I would like to strengthen 3:

4. $G$ has a non-faithful suborbit $\Lambda$ such that, if $M$ is the associated point-stabilizer, then $M^\Lambda\cong Alt(\Lambda)$ or $Sym(\Lambda)$. Here $M^\Lambda$ is the permutation group induced on $\Lambda$ by the action of $M$.

I don't know how common such groups are... So for now I'm just interested in collecting different examples to see if I can pick out some interesting properties. GAP/ MAGMA computations, as well as theoretical answers telling me what such groups might look like, are of interest to me.

Thanks in advance.

Answer 1

It is easy to find examples that satisfy conditions 1 - 3. I searched through the Atlas of Finite Simple Groups, looking for simple groups with maximal subgroups that might provide examples, where we take $G$ to be the image of the action of the simple group $S$ on the cosets of the chosen maximal subgroup $H$, and I found lots, using Magma.

In particular, I found two examples satisfying 1 - 4, where the action of the point stabilizer on the non-faithful orbit is $S_3$. These were:

1. $S={\rm PSL}(2,11)$, $H = D_{12}$.

2. $S={\rm PSL}(2,13)$, $H = D_{12}$.

But I don't think there are any more examples of that type, and I suspect that there are not many examples altogether. There are papers classifying primitive permutation groups with suborbits of length at most $5$, which you could find by searching. For length $5$, this is done in a paper by Fawcett, Giudici, Li, and Praeger, and I checked that there are no examples.

Since you asked for details of computations, here is the Magma computation for the Example 1.

> G:=PSL(2,11);                            
> M:=[m`subgroup:m in MaximalSubgroups(G)];
> [#m: m in M];
[ 55, 12, 60, 60 ]
> P:=CosetImage(G,M[2]);
> S:=Stabiliser(P,1);             
> O:=Orbits(S);
> [#o: o in O];
[ 1, 3, 3, 6, 6, 6, 6, 12, 12 ]
> [#OrbitImage(S,o): o in O];
[ 1, 6, 6, 12, 12, 12, 12, 12, 12 ]