A Weyl group $W$ for a finite dimensional Lie algebra with root system $\Phi$ must be finite and a subgroup of the symmetric group on $k = |\Phi|$ elements. Are there any further conditions $W$ must satisfy?
Specifically, what finite groups can be realised as Weyl groups?
Yes, Weyl groups must satisfy further conditions. Recall that all finite groups arise as subgroups of the symmetric group $S_n$ for some $n$. For Weyl groups, the possibilities are much more restricted. We have the Weyl groups of type $A_n,B_n,C_n,D_n$, see for example here, and the exceptional types, see here. The Weyl groups of the classical Lie algebras are given by $S_n$ and semidirect products of $S_n$ with $\mathbb{Z}_2^n$ or $\mathbb{Z}_2^{n-1}$.
