Conjugacy Classes of the Group Extension 27: G2(2)

• Author(s): Sikolia Murunga Jacinta ; Lucy Chikamai ; Vincent Marani
• Paper ID: 1706108
• Page: 12-17
• Published Date: 02-08-2024
• 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 group extension 27: G2(2), a maximal subgroup of both the symplectic group Sp8(2) and the automorphism group of the sporadic Fischer group Fi22. Using the Fischer-Clifford matrix technique and computational methods, we determine and analyze the conjugacy classes of 27: G2(2), their sizes, and centralizer orders. Our results reveal that 27: G2(2) has 111 conjugacy classes, with sizes ranging from 1 to 1,548,288. We construct Fischer-Clifford matrices for each conjugacy class representative in G2(2) and establish a fusion map from 27: G2(2) into Sp8(2). The analysis shows a strong influence of the normal subgroup 27 on the overall group structure, with many class sizes being powers of 2. We also compare the conjugacy class structure of 27: G2(2) with related groups, highlighting its unique features. This comprehensive analysis contributes to the broader understanding of group extensions, their relationship to larger structures like symplectic and sporadic groups, and lays the groundwork for future studies on the representations and character theory of 27: G2(2).

Keywords

Conjugacy Classes, Group Extension

Citations

IRE Journals:
Sikolia Murunga Jacinta , Lucy Chikamai , Vincent Marani "Conjugacy Classes of the Group Extension 27: G2(2)" Iconic Research And Engineering Journals Volume 8 Issue 2 2024 Page 12-17

IEEE:
Sikolia Murunga Jacinta , Lucy Chikamai , Vincent Marani "Conjugacy Classes of the Group Extension 27: G2(2)" Iconic Research And Engineering Journals, 8(2)