Symmetric generation has provided concise ways of constructing many classical and sporadic groups, in fact every non-abelian finite simple groups arises in this manner. A symmetric presentation for a group is a homomorphism from a progenitor, p*m : N. onto the group. We give details of our program which constructs all permutation images of a given progenitor. In a monomial progenitor, p*m :m N, the control subgroup N has a monomial action on the symmetric generators of order p > 3. We study monomial progenitors in which the control subgroup has an irreducible monomial representation, as several such progenitors have beell found to map onto sporadic groups. We classify all irreducible monomial representations of the alternating, symmetric and sporadic groups and their covering groups. We use the irreducible monomial representations of the covers of the alternating groups to construct monomial progenitors and we obtain sporadic images of several of these progenitors.