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
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.