Conjugacy Classes of the Maximal Subgroup 28: G2(2) in O+_10(2)

• Author(s): Khaemba Samuel Wafula ; Lucy Walingo Chikamai ; Vincent Marani
• Paper ID: 1706170
• Page: 406-411
• Published Date: 16-08-2024
• Published In: Iconic Research And Engineering Journals
• Publisher: IRE Journals
• e-ISSN: 2456-8880
• Volume/Issue: Volume 8 Issue 2 August-2024

Abstract

This study investigates the conjugacy class structure of the maximal subgroup 28: G2(2) within the orthogonal group O+_10(2). Using a combination of theoretical techniques from group extension theory and Clifford theory, along with computational methods implemented in GAP and Magma, we determine and classify all conjugacy classes of 28: G2(2). Our analysis reveals a total of 82 distinct conjugacy classes, significantly more than the 16 classes of G2(2) alone, reflecting the complex structure introduced by the 28 normal subgroup. We observe intricate fusion patterns from G2(2) to 28: G2(2), with some classes splitting and others lifting directly. Notably, we identify 57 new classes unique to the extension, showcasing the non-trivial interaction between 28 and G2(2). The centralizer structures of class representatives are analyzed, revealing a rich subgroup landscape within 28: G2(2). We also compare our results with the known conjugacy class structure of O+_10(2), providing insights into the embedding of 28: G2(2) as a maximal subgroup. This study contributes to the understanding of group extensions involving exceptional groups and classical groups, and lays groundwork for future investigations into the representation theory of 28: G2(2). Our findings have potential applications in coding theory, algebraic geometry, and cryptography.

Keywords

Conjugacy Classes, the Maximal Subgroup

Citations

IRE Journals:
Khaemba Samuel Wafula , Lucy Walingo Chikamai , Vincent Marani "Conjugacy Classes of the Maximal Subgroup 28: G2(2) in O+_10(2)" Iconic Research And Engineering Journals Volume 8 Issue 2 2024 Page 406-411

IEEE:
Khaemba Samuel Wafula , Lucy Walingo Chikamai , Vincent Marani "Conjugacy Classes of the Maximal Subgroup 28: G2(2) in O+_10(2)" Iconic Research And Engineering Journals, 8(2)