A reflection group is a group of transformations on a vector space generated by reflections. Many objects in modern algebra have some connection to reflection groups and this makes their study important. To a real reflection group we can associate another group, called its "cactus group”. This group permutes the elements of the reflection group and very little is known about this phenomena in examples. This project would aim to make calculations in some specific cases. Another direction for the project would be to look at certain combinatorial objects called "crystals” which also come with a relationship to the cactus group.