OD-characterization of Almost Simple Groups Related to displaystyle D4(4)

Let $G$ be a finite group and $pi_{e}(G)$ be the set of orders of all elements in $G$. The set $pi_{e}(G)$ determines the prime graph (or Grunberg-Kegel graph) $Gamma(G)$ whose vertex set is $pi(G)$, the set of primes dividing the order of $G$, and two vertices $p$ and $q$ are adjacent if and only if $pqinpi_{e}(G)$. The degree $deg(p)$ of a vertex $pin pi(G)$, is the number of edges incident on $p$. Let $pi(G)={p_{1},p_{2},...,p_{k}}$ with $p_{1}<p_{2}<...<p_{k}$. We define $D(G):=(deg(p_{1}),deg(p_{2}),...,deg(p_{k}))$, which is called the degree pattern of $G$. The group $G$ is called $k$-fold OD-characterizable if there exist exactly $k$ non-isomorphic groups $M$ satisfying conditions $|G|=|M|$ and $D(G)=D(M)$. Usually a 1-fold OD-characterizable group is simply called OD-characterizable. In this paper, we classify all finite groups with the same order and degree pattern as an almost simple groups related to $D_{4}(4)$.