OD-characterization of S4(4) and its group of automorphisms

Let G be a nite group and (G) be the set of all prime divisors of jGj. The prime graph of G is a simple graph 􀀀(G) with vertex set (G) and two distinct vertices p and q in (G) are adjacent by an edge if an only if G has an element of order pq. In this case, we write p q. Let jGj = p1 1 p2 2 pk k , where p1 < p2 < : : : < pk are primes. For p 2 (G), let deg(p) = jfq 2 (G)jp qgj be the degree of p in the graph 􀀀(G), we dene D(G) = (deg(p1); deg(p2); : : : ; deg(pk)) and call it the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups S such that jGj = jSj and D(G) = D(S). Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Let L = S4(4) be the projective symplectic group in dimension 4 over a eld with 4 elements. In this article, we classify groups with the same order and degree pattern as an almost simple group related to L. Since Aut(L) = Z4 hence almost simple groups related to L are L, L : 2 or L : 4. In fact, we prove that L, L : 2 and L : 4 are OD-characterizable.