This project is in the area of group theory and the representation theory of finite groups. Groups may be understood as collections of symmetries, whose study was motivated by the desire to understand the symmetry of an object, whether it be in nature, art, communication networks, or any other place that symmetry might play a role. Group theory has applications in physics, chemistry, and other natural sciences. In recent years, research in group theory has had a significant impact on technological advances, such as in cryptography and coding theory. Representation theory is a tool used to better understand the structure of a group and the symmetries it represents by providing a way to view an abstract group as a group of matrices, whose structure is often easier to understand. This project focuses on a number of problems which seek to relate the representation theory of a finite group to the structure and representations of certain so-called local subgroups, which reflect numerical information encoded by the group. The award will also support educational activities centering graduate student mentorship and undergraduate research. These activities include local workshops for Colorado-area graduate students and postdocs and a Directed Reading Program. The latter provides a "research-like" experience for undergraduates, in which undergraduate mentees and graduate mentors work under the guidance of the PI to study a topic outside of the students’ normal coursework. The activities