The degree pattern of a ¯nite group G was introduced in [10]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic ¯nite groups with the same order and same degree pattern as G. When a group G is 1-fold OD-characterizable, we simply call it OD-characterizable. In recent years, a number of authors attempt to characterize ¯nite groups by their order and degree pattern. In this article, we ¯rst show that for the primes p = 53, 61, 67, 73, 79, 83, 89, 97, the alternating groups Ap+3 are OD-characterizable, while the symmetric groups Sp+3 are 3-fold OD-characterizable. Next, we show that the automorphism groups Aut(O7(3)) and Aut(S6(3)) are 6-fold OD- characterizable. It is worth mentioning that the prime graphs associated with all these groups are connected.
