Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω?
Motivation: In some cases, there are so many kinds of orbits of G on Ω that every subgroup of G appears as a point-stabilizer. However, these actions are quite rare, even in the case of a point group of a group of Euclidean isometries. (The only such actions in 2D are C1, D1, and Cp for primes p. This follows because rotation subgroups only stabilize the origin.)
If we allow the group to act on subsets too, then we may have better luck. See for instance, the lovely example of dihedral groups and cyclic groups acting naturally on the Euclidean plane as Rosette groups. Here one can take various natural subsets of the polygon to get all subgroups.
In general a finite point group of a group of Euclidean isometries has a regular orbit, and the blocks of that orbit suffice to specify all subgroups (there is a 1-1 correspondence between subgroups containing a point stabilizer and blocks of the orbit given, and this correspondence is given by the set-wise stabilizer). This is true more generally for small groups acting on large vector spaces (if V is n-dimensional over K and |G|−1 < 1+|K|+…+|K|n−1, then G has a regular orbit on V in any faithful, linear action of G on V; in particular finite groups acting on infinite vector spaces are fine).
However, determining which medium-ish linear groups over medium-ish vector spaces have regular orbits is really hard, as far as I know. I wondered if it was much easier if we allowed one to replace V by V ⊕ V. Certainly we can replace V by VV (taking d=|V|), but that is a bit extreme.
Is there some reasonable value of d that can be read from the permutation character of G on Ω, from the (Brauer) character of G on V, or some other natural invariant of the action of G on Ω?
An expression for the minimum would be nice, but even some decent bound on d in terms of invariants would be fine.
To identify Galois groups, one also looks for certain invariants for polynomials, and I believe what I am asking is relevant to how complicated such invariants need to be. GAP for instance takes d ≤ 5, I think (and used to take d ≤ 3, I think), but it has lots of "backup" routines to identify various other subgroups, possibly for efficiency, but possibly because one cannot choose d that small in general.