Let (?) be a p-group of order p~n,its commutator group D(?) beingcyclic and belonging to the center(?),of(?) Suppose that the orderof D(?)is p~m,and D(?)=(t) where t is a generating element ofD(?).It is proved that the factor group(?)is an Abelian group oftype{p~(m1),p~(m1),p~(m2),p~(m2),...,P~(mr),p~(mr)} where m=m_1≧m_2≧...≧m_r.The order of(?) is p~(2r),r=m_1+m_2+m_r.(?) is the product of its sub-groups (?)When i≠j,elements of (?) commute with elements of (?) and (?)The group (?) (i=1,2,...,r) is defined as(?)the commutator of (?) and(?)Any element X of (?) can be uniquely represented asSuch elements (?) are called basis of (?)The transformation of two bases can be represented by a matrixof (?) where (?) is the set of all 2n-rowed square matrices with integralelements.It is proved that the necessary and sufficient condition for the matrix T to represent such a transformation is that TPT~0≡P(modp~m),where P=diag{p~(m-m_1),p~(m-m_1),p~(m-m_2),p~(m-m_2),...ρ~(m-m_r),ρ~(m-m_r)},and θ be atransformationθ:T←→T~0,(T and T~0 in (?))of(?)specially defined such that(A+B)~0=A~0+B~0,(AB)~0=B~0 A~0,(A~0)~0=A for every A and B of(?) The totality of such matrices Tform a group.