Let \((X,x)=\{f=0\}\subset\mathbb{C}^ 3\) be an isolated quasi-homogeneous hypersurface singularity and \(G=\langle(e_ n^{i_ 0},e_ n^{i_ 1},e_ 2^{i_ 2})\rangle\) a cyclic group which is diagonally acting on \((X,x)\), where \(e_ n=\exp(2\pi\sqrt{-1}/n)\). Then the quotient \((x/G,\pi(0))\) is a normal surface singularity with \(\mathbb{C}^*\)-action. In our paper we study those surface singularities. Although they are very special normal surface singularities with \(\mathbb{C}^*\)-action, it is meaningful to study them. Because, if we consider a singularity of such type, we can easily compute the genus \(p_ g\) and determine if it is Gorenstein, though it is not easy to do them for general normal surface singularities with \(\mathbb{C}^*\)-action.
In section 2, we give a method to resolve them; it is obtained by the techniques of P. Orlik, Ph. Wagreich and A. Fujiki. Although the algorithm is not so simple, it is easy to program for the computer. In section 3, we give a formula to compute the geometric genus \(p_ g(X/G,\pi(0))\) and a criterion for \((X/G,\pi(0))\) to be Gorenstein. In section 4, we classify all rational singularities which are obtained as cyclic quotients of the simple elliptic singularity \(\tilde E_ 8\) by a reflection free finite cyclic group \(G\). These singularities are already found as weighted dual graph from other different points of view. Our result gives concrete representations for them.