A theorem of Chevalley, Shepard, and Todd states that if $G$ is a finite group and $\rho: G \rightarrow GL\_n(\mathbb{C})$ a representation so that $\rho(G)$ is generated by pseudoreflections, then $\mathbb{C}[z\_1,\dots,z\_n]^G$ (the subring of $G$-invariant polynomials) is again a polynomial ring.


From my understanding, a pseudoreflection is diagonalizable with diagonal form $\text{diag}(1,\dots,1,\zeta)$ where $\zeta$ is a root of unity.


My question is whether the property of being generated by pseudoreflections is a property of the representation or of the group itself, and if so what is the equivalent abstract group-theoretic property. If it is not the case, could someone point me towards a group which furnishes a counterexample? 



Finite groups generated by pseudo-reflections
It's a property of the representation. For example, under the 1-dimensional representation of $\newcommand{\Z}{\mathbb Z} \Z/2$ given by $x\mapsto -x$, $\Z/2$ is generated by pseudoreflections, but under the 2-dimensional representation $(x,y)\mapsto(-x,-y)$, it's not.


It makes sense to ask if there is an abstract finite group $G$ which *cannot* be generated by pseudoreflections (i.e. there is no faithful representation under which it is generated by pseudoreflections). I feel like I should know the answer (I even feel like I've thought about this question before), but I don't.


