The circular division of the factor group (Q2r x Cr, Z)/(P)over-bar(Qr x Cr) when r is an odd number

The benefit of this paper is to find the circular divison of the factor group (Q(2r) x C-r, Z)/(P) over bar (Q(r) x C-r) when r is an odd number, where Q(2r) is denoted to Quaternion group of order 4r, such that for each positive integer r, there are two generators a and b for Q(2r) satisfies Q(2r) = {a(h) b(k) , 0 <= h <= 2r - 1, k = 0, 1} which has the following properties {a (2r) = b (4) = I, ba'b(-1) = a(-r) }.