Consider the group , by which I mean quotient of by its center. So this group has order . If an interested reader looks at the classification of groups of order :”apparently”, says this reader, “this group is isomorphic to !”
Now naturally acts on a set of 5 elements. On the other hand, naturally acts on a set of 6 elements, namely the points of .
The brain teaser is: how to find a natural action of on a set of 6 elements, and how to find a natural action of on a set of 5 elements? I couldn’t find these in a nice way. For instance, we can let act on (the projectivization of) degree 4 homogeneous polynomials (with coefficients in ). The orbit of consists of elements (okay, let me not prove this here). So this must be it. But this is not satisfying. So the brain teaser is really: how can you realize these actions in a nice way?