A Note on Embeddings of $S_4$ and $A_5$ into the Two-dimensional Cremona Group and Versal Galois Covers

Abstract

In this article, we prove that two versal Galois covers for $S_4$ and $A_5$ introduced in [17], [18] and [19] are birationally distinct to each other. As a corollary, we obtain two non-conjugate embeddings of $S_4$ and $A_5$ into ${\mathrm{Cr}}_2(C)$.