Exclusions of smooth actions on spheres of the non-split extension of $C_2$ by $\operatorname{SL}(2,5)$

Abstract

There are four groups $G$ fitting into a short exact sequence $1\rightarrow\operatorname{SL}(2,5)\rightarrow G\rightarrow C_2\rightarrow 1,$ where $\operatorname{SL}(2,5)$ is the special linear group of $(2\times 2)$-matrices with entries in the field of five elements. Except for the direct product of $\operatorname{SL}(2,5)$ and $C_2$, there are two other semidirect products of these two groups and just one non-semidirect product $\operatorname{SL}(2,5).C_2$, considered in this paper. It is known that each finite nonsolvable group can act on spheres with arbitrary positive number of fixed points. Clearly, $\operatorname{SL}(2,5).C_2$ is a nonsolvable group. Moreover, it turns out that $\operatorname{SL}(2,5).C_2$ possesses a free representation and as such, can potentially act pseudofreely with nonempty fixed point set on manifolds of arbitrarily large dimension. We prove that $\operatorname{SL}(2,5).C_2$ cannot act effectively with odd number of fixed points on homology spheres of dimensions less than $14$. In the special case of effective one fixed point actions on homology spheres, we are able to exclude $15$, $16$, and $17$ from the dimension of them. Moreover, we prove that $5$-pseudofree one fixed point actions of $\operatorname{SL}(2,5).C_2$ on spheres do not exist.