In this thesis we have investigated permutation and monomial progenitors of the form p^*n:N (p=2,3,5,...) and p^*n:_m N (p=3,5,7,...) respectively. We have discovered new symmetric presentations of several finite nonabelian simple groups including linear groups, unitary groups, orthogonal groups, and sporadic groups. We have constructed interesting groups found using the technique of double coset enumeration and found the isomorphic types of the numerous groups that appeared as homorphic images. These include the sympletic group, S_4(5) and the Janko groups, J_2 and J_1 which were found using a variety of different control groups over finite fields.