This book is a sequel to the Atlas of residually weakly primitive geometries for small groups by F. Buekenhout, M. Dehon and D. Leemans, published in 1999. The work deals with group theory, incidence geometries and computational algebra developed to better understand the structure of sporadic groups. The author describes several Magma algorithms that he has applied to classify the geometries in the title for the five Mathieu groups, the first three Janko groups and the Higman-Sims group. Then he proceeds to present some results obtained by studying the list of geometries obtained with the help of these programs. There is also a section presenting full subgroup lattices of the Mathieu groups M22 and M23 and the Janko groups J2 and J3. The book will be of great value for anyone studying finite incidence geometries and the structure of sporadic groups.