Some Quantitative Characterizations of Certain Symplectic Groups

Given a finite group $G$, denote by $\mathrm{D}(G)$ the degree pattern of $G$ and by $\mathrm{O}\mathrm{C}(G)$ the set of all order components of $G$. Denote by $h_{{\rm OD}}(G)$(resp\mathrm{.}$h_{{\rm OC}}(G)$) the number of isomorphism classes of finite groups $H$ satisfying conditions $\mathrm{\mid }H\mathrm{\mid }=\mathrm{\mid }G\mathrm{\mid }$ and $\mathrm{D}(H)=\mathrm{D}(G)$ (resp. $\mathrm{O}\mathrm{C}(H)=\mathrm{O}\mathrm{C}(G)$). A finite group $G$ is called OD-characterizable (resp. OC-characterizable) if $h_{\mathrm{O}\mathrm{D}}(G)=1$ (resp. $h_{\mathrm{O}\mathrm{C}}(G)=1$). Let $C=C_p(2)$ be a symplectic group over binary field, for which $2^p-1>7$ is a Mersenne prime. The aim of this article is to prove that $h_{\rm OD}©=1=h_{\rm OC}©$.