A finite group $G$ has a finite set of irreducible representations over the complex numbers.  All of these representations are linear (that is, are maps in 1x1 complex matrices) if and only if $G$ is abelian.  Moreover, if the group $G$ is not abelian, those representations which *are* linear can be described by replacing $G$ by $G/G'$ (where $G'$ is the commutator subgroup.)

I need a similar classification in which the field of complex numbers is replaced by the quaternion division ring.  Is there a similar theory?  **Can I classify those $G$ for which the quaternionic representations are linear?**  (Is it possible to even identify a normal subgroup $K$ of $G$ for which I am guaranteed that $G/K$ is "quaternionic linear"?)

I've identified some articles on "[quaternionic][1]" or "symplectic" representations but most of these are concerned with infinite groups and assume quite a lot of theory. 


  [1]: http://en.wikipedia.org/wiki/Quaternionic_representation