Abstract A progenitor developed by Robert T. Curtis is a type of infinite groups formed by the semi-direct product of a free group m∗n and a transitive permutation group of degree n. To produce finite homomorphic images we had to add relations to the progenitor of the form 2∗n : N. In this thesis we have investigated several permutations progenitors and monomials, 2∗12 : S4, 2∗12 : S4 × 2, 2∗13 : (13 : 4), 2∗30 : ((2• : 3) : 5), 2∗13 :13,2∗13 :(13:2),2∗13 :(13:S3),53∗2 :m (13:4),7∗8 :m (32 :8),and 53∗4 :m (13 : 4). We have discovered that the permutations progenitors produced the following finite homomorphic images, we have found P GL(2, 13), U3 (4) : 2, 2 × Sz (8), PSL(2,7), PGL(2,27), PSL(2,8), PSL(3,3), 4•S4(5), PSL2(53), and 13 : PGL2(53) as homomorphic images of this progenitors. We will construct double coset enumeration for the homomorphic images, 2 × Sz (8) over (13 : 4) Suzuki twisted group, P GL(2, 13) over S4,and PSL(2,7) over S4 and Maximal subgroups of 2×PGL(2,27) over 2•(13 : 2), P SL(2, 8) over (9 : 2), and P SL(3, 3) over (13 : 3). We will also give our techniques of finding finite homomorphic images and their isomorphism images.