Error-correcting codes with large automorphism groups are valuable for studying the properties of codes and designing efficient decoding algorithms. There are various approaches to obtain linear codes from the primitive permutation representations of finite simple groups. The aim of this project is to construct linear codes arising from finite primitive groups. I mainly study these codes from a modular representation theoretic perspective. While I use Magma for computational approaches, a strong theoretical framework is often required to handle cases where the groups are too large for computation. I also explore applications of coding theory in cryptography, secret sharing schemes, and related areas. 

This project focuses on constructing combinatorial designs from the maximal subgroups and conjugacy classes of finite groups. By utilising Key-Moori methods and their generalisations, we aim to develop designs through primitive permutation representations. While previous work has addressed certain families of simple groups, ongoing efforts are extending these constructions to include orthogonal groups, unitary groups, alternating groups, and some generalisations of the method. 

There are several types of graphs whose vertex sets are related to a group G, and whose edges in some way reflect the properties of the group. The vertex set may consist of the elements of G, the element orders, the conjugacy classes of G, or the irreducible character degrees. In this project, we study the commuting conjugacy class graph of a group, where the vertex set is the set of non-identity conjugacy classes of G, and two conjugacy classes C and D are joined by an edge if and only if there exist x in C and y in D such that [x, y] = 1. There is still much that remains unknown about these graphs. 

This project explores impartial combinatorial games played on finite groups, with a focus on permutation groups. In these games, two players alternately select elements from a group or a related subset, and the outcome is determined by algebraic properties of the selections. A central goal is to determine Sprague–Grundy numbers (nimbers) for these games, which encode winning strategies and outcomes. The research connects combinatorial game theory with group theory, investigating how the structure of a group influences the play and outcome of games. Applications include generating sets, avoidance and achievement games, and understanding strategic interactions constrained by group structure.