{"id":5215,"date":"2025-06-25T11:45:17","date_gmt":"2025-06-25T11:45:17","guid":{"rendered":"https:\/\/fs.kibu.ac.ke\/?p=5215"},"modified":"2025-12-26T22:56:32","modified_gmt":"2025-12-26T22:56:32","slug":"conjugacy-classes-of-the-group-27-g22","status":"publish","type":"post","link":"https:\/\/kibu.ac.ke\/fs\/conjugacy-classes-of-the-group-27-g22\/","title":{"rendered":"Conjugacy Classes of the Group 27: G2(2)"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"5215\" class=\"elementor elementor-5215\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-798b43d1 elementor-section-full_width elementor-section-height-min-height elementor-section-items-stretch elementor-section-height-default\" data-id=\"798b43d1\" data-element_type=\"section\" id=\"ourgoal\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6cd7e3b9\" data-id=\"6cd7e3b9\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-53915b13 elementor-widget elementor-widget-theme-post-title elementor-page-title elementor-widget-heading\" data-id=\"53915b13\" data-element_type=\"widget\" data-widget_type=\"theme-post-title.default\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Conjugacy Classes of the Group 27: G2(2)<\/h1>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-34fa58d0 elementor-section-height-min-height elementor-section-boxed elementor-section-height-default elementor-section-items-middle\" data-id=\"34fa58d0\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-33a9ce1a\" data-id=\"33a9ce1a\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-f0701bc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f0701bc\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-2d0d41a8 elementor-invisible\" data-id=\"2d0d41a8\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-35f3fa40 elementor-widget__width-auto elementor-widget elementor-widget-heading\" data-id=\"35f3fa40\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">2024<\/h2>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-3bf85be6 elementor-invisible\" data-id=\"3bf85be6\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-396155d0 elementor-widget elementor-widget-heading\" data-id=\"396155d0\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">KIBU Authors<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4c49cf14 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"4c49cf14\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeIn&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p>Lucy Chikamai<\/p><p>Shem Aywa<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-5b5e3c8a elementor-invisible\" data-id=\"5b5e3c8a\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-45f8b02f elementor-tablet-align-left elementor-invisible elementor-widget elementor-widget-button\" data-id=\"45f8b02f\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInLeft&quot;}\" data-widget_type=\"button.default\">\n\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-shrink\" href=\"https:\/\/www.irejournals.com\/paper-details\/1706129\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">VIEW ON PUBLISHER SITE<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-46415076 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"46415076\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-inner-column elementor-element elementor-element-2edf6392 elementor-invisible\" data-id=\"2edf6392\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5bf18449 elementor-widget elementor-widget-heading\" data-id=\"5bf18449\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Abstract<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6021cf72 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"6021cf72\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeIn&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p>The group 27: G2(2), a maximal subgroup of the automorphism group of the Fischer group Fi22, is a semi-direct product of an elementary abelian group of order 27 and the exceptional group G2(2). This study aims to comprehensively determine and analyze the conjugacy classes of 27: G2(2) using computational methods. The conjugacy classes of a group provide valuable insights into its structure, symmetry, and representation theory, and have potential applications in various areas of mathematics, such as coding theory, cryptography, and mathematical physics. Using the computer algebra systems GAP and MAGMA, we computed the complete set of conjugacy classes of 27: G2(2) and analyzed their structure, sizes, and power maps. The results reveal that 27: G2(2) has 60 conjugacy classes, with sizes ranging from 1 to 110,592, and a non-uniform distribution of classes by element order. Comparison with the conjugacy classes of G2(2) shows that the extension by 27 leads to a splitting of conjugacy classes, with varying splitting patterns for different element orders. This study contributes significantly to the understanding of the structure of 27: G2(2) and lays the foundation for further exploration of its properties and applications. The conjugacy class information obtained in this research can be used in the study of representation theory, coding theory, cryptography, and mathematical physics. Our findings also provide insights into the relationship between 27: G2(2), its constituent group G2(2), and the larger context of the Fischer group Fi22 and its automorphism group.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>2024 KIBU Authors Lucy Chikamai Shem Aywa VIEW ON PUBLISHER SITE Abstract The group 27: G2(2), a maximal subgroup of the automorphism group of the Fischer group Fi22, is a semi-direct product of an elementary abelian group of order 27 and the exceptional group G2(2). This study aims to comprehensively determine and analyze the conjugacy [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_header_footer","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"class_list":["post-5215","post","type-post","status-publish","format-standard","hentry","category-research-list"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Conjugacy Classes of the Group 27: G2(2) - Faculty of Science<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/kibu.ac.ke\/fs\/conjugacy-classes-of-the-group-27-g22\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Conjugacy Classes of the Group 27: G2(2) - Faculty of Science\" \/>\n<meta property=\"og:description\" content=\"2024 KIBU Authors Lucy Chikamai Shem Aywa VIEW ON PUBLISHER SITE Abstract The group 27: G2(2), a maximal subgroup of the automorphism group of the Fischer group Fi22, is a semi-direct product of an elementary abelian group of order 27 and the exceptional group G2(2). This study aims to comprehensively determine and analyze the conjugacy [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/kibu.ac.ke\/fs\/conjugacy-classes-of-the-group-27-g22\/\" \/>\n<meta property=\"og:site_name\" content=\"Faculty of Science\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-25T11:45:17+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-12-26T22:56:32+00:00\" \/>\n<meta name=\"author\" content=\"kibabii\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"kibabii\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/conjugacy-classes-of-the-group-27-g22\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/conjugacy-classes-of-the-group-27-g22\/\"},\"author\":{\"name\":\"kibabii\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#\/schema\/person\/fc9b7f13eeb65cf0be520197c35b17d6\"},\"headline\":\"Conjugacy Classes of the Group 27: G2(2)\",\"datePublished\":\"2025-06-25T11:45:17+00:00\",\"dateModified\":\"2025-12-26T22:56:32+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/conjugacy-classes-of-the-group-27-g22\/\"},\"wordCount\":246,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#organization\"},\"articleSection\":[\"Research List\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/kibu.ac.ke\/fs\/conjugacy-classes-of-the-group-27-g22\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/conjugacy-classes-of-the-group-27-g22\/\",\"url\":\"https:\/\/kibu.ac.ke\/fs\/conjugacy-classes-of-the-group-27-g22\/\",\"name\":\"Conjugacy Classes of the Group 27: G2(2) - 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