Simple singularity of type $E_7$ and the complex reflection group ST34

This paper studies a family of surfaces of ${\mathbf{C}}^3$ which is a deformation of a simple singularity of type $E_7$. This family has six parameters which are regarded as basic invariants of the complex reflection group No.34 in the list of the paper of Shephard and Todd \cite{ST}. We compute 1-parameter subfamilies of the family in question corresponding to corank one reflection subgroups of No.34 group. In particular, we determine the types of simple singularities on the surfaces appeared in this manner.