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

Let G be a finite group and ?e(G) be the set of orders of all elements in G. The set ?e(G) determines the prime graph (or GrunbergKegel graph) ?(G) whose vertex set is ?(G). The set of primes dividing the order of G, and two vertices p and q are adjacent if and only if pq ? ?e(G). The degree deg(p) of a vertex p ? ?(G), is the number of edges incident on p. Let ?(G) = {p1, p2, ..., pk} with p1 < p2 < ... < pk. We define D(G) := (deg(p1), deg(p2), ..., deg(pk)), 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 D4(4).