TITLE: Which finite nonabelian groups have all their quaternionic representations of degree one?
QUESTION [10 upvotes]: 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" or "symplectic" representations but most of these are concerned with infinite groups and assume quite a lot of theory that does not seem relevant to finite groups.
(This mathoverflow question is similar but in that post $G$ is infinite and there is not the emphasis on "all representations are degree 1".)

REPLY [7 votes]: The original question has been answered. As background information, I point out that S. Amitsur classified finite groups which can occur as subgroups of multiplicative groups of divison algebras. These include finite groups which have a faithful "one-dimensional" representation over the quaternions. As noted in Ben's answer, any such finite group (in the general division ring case) has cyclic Sylow $p$-subgroups for each odd prime $p$ and cyclic or generalized quaternion Sylow $2$-subgroups. It also satisfies other group-theoretic properties, such as elements of order $p$ and $q$ commuting when $p$ and $q$ are distinct primes. 
In fact, finite groups which occur as subgroups of multiplicative groups of division algebras are Frobenius complements. We note that any generalized quaternion group already occurs - this is not in conflict with Ben's answer: if that group has order greater than $8,$ then it has a non-Abelian dihedral group as a homomorphic image, so not all its irreducible representations over the quaternions are $1$-dimensional. The only non-solvable group which occurs is ${\rm SL}(2,5),$ which does already occur inside the multiplicative group of the real quaternions ( as does each generalized quaternion $2$-group)- again, the fact that ${\rm SL}(2,5)$ has $A_{5}$ as a homomorphic image means that not all its irreducible representations over the quaternions are $1$-dimensional.