We investigate the terminal symplectic partial resolutions of quotients of C 4 by exceptional Shephard-Todd groups. We describe all terminal symplectic partial resolutions of C4/G4. For the quotient C4/G5 we find a new isolated conical symplectic singularity ÿ, which is 4-dimensional, isolated, and locally simply-connected. We prove ÿ is distinguished from the other such singularities by the fact that its tangent cone at the singular point is non-reduced. We then give a method of finding a terminal symplectic partial resolution by its contraction, and demonstrate it to find all such for C4/G6. We describe the singularities present in all Q-factorial terminalisations of C4/G7. We describe the unique terminal symplectic partial resolution of C4/G12. Finally, motivated by the results of C4/G12, we detail a method to find quotients of terminal symplectic singularities which are Q-factorial.
Callum Berry (Fri,) studied this question.