Orders of Automorphism Groups of Some p-Groups

The orders of automorphism groups of the groups of order p6 in the twelve family φ12 are produced, where p is an odd prime. Every group is analysed by utilizing the properties of metabelian, regularity and p-commutativity of finite p-groups, and the structure of the generators of its automorphism groups is obtained. Then the orders of automorphism groups are determined through some properties of equivalence in number theory.