<b>Background:</b>

Let $K$ be a field and let $V$ be a finite-dimensional $K$-vector space. A _pseudoreflection_ (or usually imprecisely just _reflection_) in $V$ is an element $1 \neq s \in \mathrm{GL}(V)$ fixing a hyperplane. A _reflection representation_ of a group $W$ over $K$ is a $K$-linear representation $\rho:W \rightarrow \mathrm{GL}(V)$, such that $\rho(W)$ is generated by reflections. A group $W$ is called a _reflection group_ over $K$ if it admits a reflection representation over $K$.

Shephard-Todd classified the finite irreducible reflection groups over $\mathbb{C}$ (i.e. those finite groups admitting an irreducible reflection representation over $\mathbb{C}$). 

<b>Question:</b>

Is there also a classification of the finite irreducible reflection representations over $\mathbb{C}$? [One could say that two reflection representations $\rho:G \rightarrow \mathrm{GL}(V)$, $\rho':G' \rightarrow \mathrm{GL}(V')$ are _isomorphic_ if there exists a vector space isomorphism $f:V \rightarrow V'$ such that $f \rho(G) f^{-1} = \rho'(G)$)]. Is it known for each of the finite irreducible reflection groups over $\mathbb{C}$ how many irreducible reflection representations they have? I'm pretty sure all this is well-known but in no book/article I've looked at this is mentioned...