Progenitors Involving Simple Groups

I will be going over writing representations of both permutation and monomial progenitors, which include 2^{*4} : D_4, 2^(*7) :L_2 (7) as permutation progenitors, and monomial progenitors 7^(*2) :_m S_3 \times 2, 11^{*2} :_m (5:2)^{*}5, 11^{*3} :_m (25:3), 11^{*4} :_m (4 : 5)^{*}5. Also, the images of these different progenitors at both lower and higher fields and orders. \\ We will also do the double coset enumeration of S5 over D6, S6 over 5 : 4, A_5 x A_5 over (5:2)^{*}5, and go on to also do the double coset enumeration over maximal subgroups for larger constructions. We will also do the construction of sporadic group M22 over maximal subgroup A7, and also J1 with the monomial representation 7^(*2) :_m S_3 \times 2 over maximal subgroup PSL(2,11). We will also look at different extension problems of composition factors of different groups, and determine the isomorphism types of each extension.