We investigate the Peterson hit problem for the polynomial algebra P d , viewed as a graded left module over the mod-2 Steenrod algebra, A. For d > 4, this problem is still unsolved, even in the case of d = 5 with the help of computers. In this article, we study the hit problem for the case d = 6 in the generic degree 6(2 r − 1) + 6.2 r , with r an arbitrary non-negative integer. Furthermore, the behavior of the sixth Singer algebraic transfer in degree 6(2 r − 1) + 6.2 r is also discussed at the end of this paper.