Quite a long time ago, Arunas Rudvalis discovered a symmetric 2-(14080,1444,148) design: a set of 14080 points, with 14080 subsets of size 1444 called blocks, with the property that any point lies in 1444 blocks, any two points in 148 blocks, and any two blocks meet in 148 points. A few years ago, when I visited Harriet Pollatsek in Mt Holyoke, I met up with Arunas, and he suggested that we delve deeper into the structure of this design, which we did, and published a paper on a remarkable geometry it conceals.
At roughly the same time, I worked with Cheryl Praeger on designs with flag-transitive but point-imprimitive automorphism groups. These are fairly rare, and always have a beautiful structure involving number-theoretic or group-theoretic coincidences. Symmetric designs are even rarer. But we were led to suspect the existence of a symmetric 2-(1408,336,80) design. (Why one with one-tenth the number of points of the Rudvalis design? I have no idea!)
This never got published. The reason was that we (well, mainly Cheryl) developed a very general construction method, extending an earlier idea by Sharad Sane. The ingredients are three designs (one symmetric, one resolvable, and one group-divisible) with parameters related in a certain way, together with some bijections with appropriate properties. Our construction of our new design (and indeed, some known designs with subgroups of their full automorphism groups which are flag-transitive but block-imprimitive) were purely group-theoretic, and to a casual glance bore no resemblance to our general methods. Indeed, it can be quite hard to say exactly what designs and bijections should be put into the general method in order to produce these designs.
I will describe here the construction of the symmetric 2-(1408,336,80) design, because I have a small apology to make.
The group 3.M22 has a 6-dimensional representation over the field GF(4), giving rise to a semi-direct product G = 212:(3.M22). (Matrices generating 3.M22 can be obtained from the on-line Atlas of Finite Group Representations, and downloaded into a GAP program.) Restricting to the subgroup 3.M21, the 6-dimensional module has a 3-dimensional submodule, and so we obtain a subgroup H = 26:(3.M21) of G. So we can represent G as a permutation group of degree 22×64=1408 on the cosets of H. (Computationally, constructing this permutation representation is by far the most time-consuming part of the exercise.)
Now G is imprimitive, with 22 blocks of size 64; the group permuting the blocks is the 3-transitive M22. The stabiliser of a point has an orbit of length 336, which meets every block except the one containing the stabilised point in 16 points. The 1408 images of this point under G are the blocks of the required design.
My apology is for claiming, in various places, that the automorphism group of this design is the group used in the construction. In fact it is twice as large (though still flag-transitive and point-imprimitive). The outer automorphism of M22 acts as a field automorphism over GF(4), so is not visible in the linear action on the 6-dimensional module; but it does preserve the design. So the full group has structure 212:((3.M22):2).
It is no coincidence that I am thinking about this while Cheryl and I are in the same town, as you will not be surprised to learn if you have ever worked with Cheryl!