Abstract In this project, we searched for new constructions and symmetric presentations of important groups, nonabelian simple groups, their automorphism groups, or groups that have these as their factor groups. My target nonabelian simple groups included sporadic groups, linear groups, and alternating groups. In addition, we discovered finite groups as homomorphic images of progenitors and proved some of their isomorphism type and original symmetric presentations. In this thesis we found original symmeric presentations of M12, J1 and the simplectic groups S(4,4) and S(3,4) on various con- trol groups. Using the technique of double coset enumeration we constucted J2 as a homomorphic image of the permutation progenitor 2∗10 : (10 × 2). From our mono- mial progenitor 11∗4 : (2 : 4) we found a homomorphic image of M11. In the following chapters we will discuss how we went about obtaining homomorphic images, some con- structions of the Cayley Diagrams, and how we solved some extension problems.