# Explicit permutation representation of the Thompson sporadic simple group?

The Thompson group Th of order $90745943887872000$ is one of the sporadic simple groups occurring in the classification of finite simple groups.

Its maximal subgroups are known (see http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Th/) and they are all remarkably small, which has the consequence that any permutation representation of Th has very large degree.

In particular, the permutation representation of Th of lowest possible degree has degree $143127000$.

Question: Has anyone explicitly determined two permutations of this degree that generate Th, and if so, are these two permutations available online?

Google reveals that some CS researchers have previously used this group as a test case for developing, refining and testing algorithms for high-degree permutation groups.

For example, the paper

http://www.ccs.neu.edu/home/gene/papers/issac03.pdf

explicitly mentions the Thompson group, and more generally algorithms for manipulating permutation groups of degree higher than 100 million. They comment that each permutation needs about 1/2 Gb to store, and that as the usual algorithms need to store $\Omega(\log n)$ permutations ($n$ is the degree) space is a major problem. But this was over ten years ago, and nowadays storing even a few hundred permutations of this size would be no particular problem.

I have emailed one of the authors of this paper (Cooperman) but after waiting a reasonable time (a week or so), I have not had a reply, so I'm now asking this list.

The following Magma code worked in less than an hour. It is using the idea suggested by Dima Pasechnik working with the group ${\rm Th} < {\rm GL}(248,2)$, acting on an orbit of a subspace of dimension $2$ fixed by the maximal subgroup $^3D_4(2):3$. It used about 43GB. The group is coming from the ATLAS database, which is available from Magma, and the definition of the function SS, which defines generators of the subgroup, was taken from here

Of course, while doing this computation, it is necessary to store the 143 million images of the $2$-dimensional subspace, which will need a lot of space. Once the two permutations have been computed, this can all be thrown away. I seem to remember from a talk in Aachen several years ago that some techniques have been developed to reduce the space needed during large orbit computations of this kind, possibly at the cost of risking errors. I don't remember the details, but someone else might!

G:=MatrixGroup("Th",1);
SS:=function(S)
// WARNING! This is not an SLP!
w1 := S.1;
w2 := S.2;
w3 := w1 * w2;
w4 := w3 * w2;
w5 := w3 * w4;
w6 := w3 * w5;
w7 := w6 * w3;
w8 := w7 * w4;
w9 := w3 * w8;
w8 := w7 * w9;
w2 := w8^5;
w4 := w3^8;
w3 := w4^-1;
w5 := w3 * w1;
w1 := w5 * w4;
return [w1,w2];
end function;
H:=sub<Generic(G)|SS(G)>;
MH:=GModule(H);
C:=Submodules(MH);
V:=C[2];  //dimension 2
f:=Morphism(V,MH);
U:=VectorSpace(MH);
W:=sub<U|U!f(V.1),U!f(V.2)>; //setting up V as a subspace of U
I:=OrbitImage(G,W);
Degree(I); // 143127000


I presume that it is an exercise to compute these permutations, provided that one can find a nice small-dimensional representation (hopefully of dimension 248 over a small field) of Th s.t. the subgroup of minimal index (the information how to generate it is given in http://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/ThG1-max1W1) stabilises some small-dimensional subspace there, and then compute the action of generators on the orbit of these subspaces.