On the Characterization of Sporadic Simple Groups by Codegrees

Let $G$ be a finite group and $\mathrm{I}\mathrm{r}\mathrm{r}(G)$ the set of all irreducible complex characters of $G$. Define the codegree of $\chi \in \mathrm{I}\mathrm{r}\mathrm{r}(G)$ as $\mathrm{c}\mathrm{o}\mathrm{d}(\chi ):=\frac{\mathrm{\mid }G:\mathrm{k}\mathrm{e}\mathrm{r}(\chi )\mathrm{\mid }}{\chi (1)}$ and denote by $\mathrm{c}\mathrm{o}\mathrm{d}(G):=\{\mathrm{c}\mathrm{o}\mathrm{d}(\chi )\mathrm{\mid }\chi \in \mathrm{I}\mathrm{r}\mathrm{r}(G)\}$ the codegree set of $G$. Let $H$ be one of the $26$ sporadic simple groups. In this paper, we show that $H$ is determined up to isomorphism by cod$(H)$.