Minimal ring of realization of irreducible representations

In characteristic zero
Suppose $$G$$ is a finite group and $$R$$ is an integral domain of characteristic zero, i.e., it contains the ring of integers as a subring. We say that $$R$$ is a minimal ring of realization of irreducible representations if all irreducible representations of $$G$$ over some splitting field containing $$R$$ can be realized with matrix entries all from $$R$$ and such that no subring of $$R$$ has this property.

Facts

 * Linear representation is realizable over principal ideal domain iff it is realizable over field of fractions: In particular, if $$R$$ is a principal ideal domain, then any representation that can be realized over the field of fractions can also be realized over $$R$$. This fact allows us to compute some minimal rings of realization.
 * Minimal ring of realization of irreducible representations need not be unique
 * Minimal rings of realization of irreducible representations for the same group may have different degrees as extensions of the ring of integers