Abstract: In 1892, Klein’s Erlangen program proposed that all geometric problems should ultimately be studied through the lens of group theory. In the 1950s, Jacques Tits introduced coset geometries, a structure that bridges geometries and their automorphism groups, allowing properties of groups to be studied via geometry and vice versa. Coset geometries play a central role in establishing the one-to-one correspondence between regular polytopes and a class of groups known as string C-groups. Consequently, classifying polytopes becomes equivalent to classifying these groups. This correspondence also makes it possible to extend classical group constructions — such as free products, HNN-extensions, and semi-direct products — into the geometric setting, where they appear as natural geometric operations. In this seminar, we will discuss the classification of regular polytopes with automorphism group which are subgroups of Sym(n), as well as these generalised geometric products. We will conclude with some possible directions for further development of this work.