Finite p-groups with a minimal non-abelian subgroup of index p (I)

For an odd prime p, we classify finite p-groups with a unique minimal non-abelian subgroup of index p. In fact, such groups have a maximal quotient which is a 3-group of maximal class. This paper is a part of classification of finite p-groups with a minimal non-abelian subgroup of index p, and partly solves a problem proposed by Berkovich.