Realizability of Two-dimensional Linear Groups over Rings of Integers of Algebraic Number Fields

Given the ring of integers OK of an algebraic number field K, for which natural numbers n there exists a finite group G ⊂ GL(n, OK) such that OKG, the OK-span of G, coincides with M(n, OK), the ring of (n × n)-matrices over OK? The answer is known if n is an odd prime. In this paper we study the case n = 2; in the cases when the answer is positive for n = 2, for n = 2m there is also a finite group G ⊂ GL(2m, OK) such that OKG = M(2m, OK).