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

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