The exact number of conjugacy classes of the Sylow p -subgroups of GL(n,q)GL(n,q) modulo (q-1)13(q-1)13

Let G be a finite p  -group of order pnpn. A well known result of P. Hall determines the number of conjugacy classes of G,r(G)G,r(G), modulo (p2-1)(p-1)(p2-1)(p-1). Namely, he proved the existence of a non-negative constant k   such that r(G)=n(p2-1)+pe+k(p2-1)(p-1)r(G)=n(p2-1)+pe+k(p2-1)(p-1).We denote by GnGn the group of the upper unitriangular matrices over FqFq, the finite field with q=ptq=pt elements. In [A. Vera-López, J. M. Arregi and F. J. Vera-López.   On the number of Conjugacy Classes of the Sylow pp-subgroups of GL(n,q)GL(n,q). Bull. Austral. Math. Soc 53,(1996), 431-439.] the number r(Gn)r(Gn) is given modulo (q-1)5(q-1)5.In this paper, we introduce the concept of primitive canonical matrix. The knowledge of the number of primitive canonical matrices with connected graph of size less than or equal to n should be sufficient to determine the number of all canonical matrices of size n  . Moreover, we give explicitly the polynomial formulas μi=μi(n),i=0,…,12μi=μi(n),i=0,…,12, depending only on n, and not on q, such thatr(Gn)=∑imui(n)(q-1)i+k(n,q)(q-1)13∀n∈N.