Abstract
In [4], Hammas and Al-Mutairi proved that Skn+r = < x, y, t > and A(kn+r) = < x, y, t > can be generated by using a copy of S-n congruent to < x, y > and an element t of order k + r, where k >= 2 is the number of disjoint cycles of the element x and 2 <= r >= n is the number of fixed points under x and y. In this paper, we use the permutation representation of the symmetric and alternating groups of degree kn + r to get presentation of these groups when k = 2 and r = 2, 3 and 4.