Quasirecognition by Prime Graph of $^2D_{n}(3^\alpha)$ where $n=4m+1\ge 21$ and $\alpha$ is Odd

Let $G$ be a finite group. The prime graph of $G$ is denoted by $\Gamma(G)$. In this paper, as the main result, we show that if $G$ is a finite group such that $\Gamma(G)=\Gamma(^2D_n(3^\alpha))$, where $n=4m+1$ and $\alpha$ is odd, then $G$ has a unique non-Abelian composition factor isomorphic to $^2D_n(3^\alpha)$. We also show that if $G$ is a finite group satisfying $|G|=|^2D_n(3^\alpha)|$, and $\Gamma(G)=\Gamma(^2D_n(3^\alpha))$, then $G\cong{}^2D_n(3^\alpha)$. As a consequence of our result, we give a new proof for a conjecture of Shi and Bi for $^2D_n(3^\alpha)$. Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered. Specifically, it is proved that $^2D_n(3^\alpha)$ is quasirecognizable by the spectrum.