Conjugacy Classes of the Split Extension 28: U4(2)

Abstract

This study presents a comprehensive analysis of the conjugacy classes of the split extension 28: U4(2), where U4(2) is the unitary group of degree 4 over the field with 2 elements. Using a combination of theoretical techniques, including Fischer-Clifford matrices and character theory, along with computational tools such as GAP and MAGMA, we determined and classified all 49 conjugacy classes of this group. Our analysis revealed complex fusion patterns from U4(2) to 28: U4(2), including class splitting and the introduction of new element orders. We found that 28: U4(2) has more than double the number of conjugacy classes compared to U4(2) alone, with class sizes ranging from 5 to over 1.3 million elements. This work addresses significant gaps in the existing literature regarding this specific group extension and provides insights into its structure, representations, and automorphisms. The methodology and results presented here contribute to the broader understanding of group extensions and lay the groundwork for further investigations into the properties and applications of 28: U4(2) in areas such as coding theory and quantum mechanics.