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<?xml version="1.0" encoding="UTF-8" ?>DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd " >html xmlns="http://www.w3.org/1999/xhtml " xml:lang="en" >head >title >GAP (CTblLibXpls) - Contents</title >meta http-equiv="content-type" content="text/html; charset=UTF-8" />meta name="generator" content="GAPDoc2HTML" />link rel="stylesheet" type="text/css" href="manual.css" />script src="manual.js" type="text/javascript" ></script >script type="text/javascript" >overwriteStyle();</script >head >body class="chap0" onload="jscontent()" >div class="chlinktop" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0.html" >Top</a> <a href="chap1.html" >1</a> <a href="chap2.html" >2</a> <a href="chap3.html" >3</a> <a href="chap4.html" >4</a> <a href="chap5.html" >5</a> <a href="chap6.html" >6</a> <a href="chap7.html" >7</a> <a href="chap8.html" >8</a> <a href="chap9.html" >9</a> <a href="chap10.html" >10</a> <a href="chap11.html" >11</a> <a href="chapBib.html" >Bib</a> <a href="chapInd.html" >Ind</a> </div >div class="chlinkprevnexttop" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap1.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap0_mj.html" >[MathJax on]</a></p>"X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5" ></a></p>div class="pcenter" >h1 >Computations with the <strong class="pkg" >GAP</strong > Character Table Library</h1 >div >br />Email: <span class="URL" ><a href="mailto:sam@math.rwth-aachen.de" >sam@math.rwth-aachen.de</a></span >br />Homepage: <span class="URL" ><a href="https://www.math.rwth-aachen.de/~Thomas.Breuer " >https://www.math.rwth-aachen.de/~Thomas.Breuer</a></span >"X81488B807F2A1CF1" name="X81488B807F2A1CF1" ></a></p>rsion 3 or later, see <span class="URL" ><a href="http://www.gnu.org/licenses " >http://www.gnu.org/licenses</a></span >.</p>"X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8" ></a></p>div class="contents" >"contents" name="contents" ></a></h3>div class="ContChap" ><a href="chap1.html#X8354C98179CDB193" >1 <span class="Heading" >Maintenance Issues for the <strong class="pkg" >GAP</strong > Character Table Library</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X7ECA800587320C2C" >1.1 <span class="Heading" >Disproving Possible Character Tables (November 2006)</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X795DCCEA7F4D187A" >1.1-1 <span class="Heading" >A Perfect Pseudo Character Table (November 2006)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X80F0B4E07B0B2277" >1.1-2 <span class="Heading" >An Error in the Character Table of <span class="SimpleMath" >E_6(2)</span > (March 2016)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7D7982CD87413F76" >1.1-3 <span class="Heading" >An Error in a Power Map of the Character Table of <span class="SimpleMath" >2.F_4(2).2</span > (November 2015)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X836E4B6184F32EF5" >1.1-4 <span class="Heading" >A Character Table with a Wrong Name (May 2017)</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X8159D79C7F071B33" >1.2 <span class="Heading" >Some finite factor groups of perfect space groups (February 2014)</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X8710D4947AEB366F" >1.2-1 <span class="Heading" >Constructing the space groups in question</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X84E7FE70843422B0" >1.2-2 <span class="Heading" >Constructing the factor groups in question</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X79109A20873E76DA" >1.2-3 <span class="Heading" >Examples with point group <span class="SimpleMath" >A_5</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X83523D1E792F9E01" >1.2-4 <span class="Heading" >Examples with point group <span class="SimpleMath" >L_3(2)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7A01A9BC846BE39A" >1.2-5 <span class="Heading" >Example with point group SL<span class="SimpleMath" >_2(7)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7D3100B58093F37D" >1.2-6 <span class="Heading" >Example with point group <span class="SimpleMath" >2^3.L_3(2)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X80800F3B7D6EF06C" >1.2-7 <span class="Heading" >Examples with point group <span class="SimpleMath" >A_6</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7D43452C79B0EAE1" >1.2-8 <span class="Heading" >Examples with point group <span class="SimpleMath" >L_2(8)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X8575CE147A9819BF" >1.2-9 <span class="Heading" >Example with point group <span class="SimpleMath" >M_11</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7C0201B77DA1682A" >1.2-10 <span class="Heading" >Example with point group <span class="SimpleMath" >U_3(3)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X85D9C329792E58F3" >1.2-11 <span class="Heading" >Examples with point group <span class="SimpleMath" >U_4(2)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X8635EE0B78A66120" >1.2-12 <span class="Heading" >A remark on one of the example groups</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X8448022280E82C52" >1.3 <span class="Heading" >Generality problems (December 2004/October 2015)</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7D1A66C3844D09B1" >1.3-1 <span class="Heading" >Listing possible generality problems</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X80EB5D827A78975A" >1.3-2 <span class="Heading" >A generality problem concerning the group <span class="SimpleMath" >J_3</span > (April 2015)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X82C37532783168AA" >1.3-3 <span class="Heading" >A generality problem concerning the group <span class="SimpleMath" >HN</span > (August 2022)</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X7D8C6D1883C9CECA" >1.4 <span class="Heading" >Brauer Tables that can be derived from Known Tables</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7DF018B77E722CA7" >1.4-1 <span class="Heading" >Brauer Tables via Construction Information</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X795419A287BD228E" >1.4-2 <span class="Heading" >Liftable Brauer Characters (May 2017)</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X864EFF897A854F89" >1.5 <span class="Heading" >Information about certain subgroups of the Monster group</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X82C7A03684DD7C6E" >1.5-1 <span class="Heading" >The Monster group does not contain subgroups of the type <span class="SimpleMath" >2.U_4(2)</span > (August 2023)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X87EC0C48866D1BDE" >1.5-2 <span class="Heading" >Perfect central extensions of <span class="SimpleMath" >L_3(4)</span > (August 2023)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap1.html#X7F605CA28441687F" >1.5-3 <span class="Heading" >The character table of <span class="SimpleMath" >(2 × O_8^+(3)).S_4 ≤ 2.B</span > (October 2023)</span ></a>span >div ></div >div >div class="ContChap" ><a href="chap2.html#X7B77FD307F0DE563" >2 <span class="Heading" >Using Table strong class="pkg" >GAP</strong ></span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X8389AD927B74BA4A" >2.1 <span class="Heading" >Overview</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7B6AEBDF7B857E2E" >2.2 <span class="Heading" >Theoretical Background</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X78EBF9BA7A34A9C2" >2.2-1 <span class="Heading" >Character Table Automorphisms</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X832525DE7AB34F16" >2.2-2 <span class="Heading" >Permutation Equivalence of Character Tables</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7906869F7F190E76" >2.2-3 <span class="Heading" >Class Fusions</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X80C37276851D5E39" >2.2-4 <span class="Heading" >Constructing Character Tables of Certain Isoclinic Groups</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7AEFFEEC84511FD0" >2.2-5 <span class="Heading" >Character Tables of Isoclinic Groups of the Structure <span class="SimpleMath" >p.G.p</span >span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X78F41D2A78E70BEE" >2.2-6 <span class="Heading" >Isoclinic Double Covers of Almost Simple Groups</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X834B42A07E98FBC6" >2.2-7 <span class="Heading" >Characters of Normal Subgroups</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X787F430E7FDB8765" >2.3 <span class="Heading" >The Constructions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X82E75B6880EC9E6C" >2.3-1 <span class="Heading" >Character Tables of Groups of the Structure <span class="SimpleMath" >M.G.A</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7CCABDDE864E6300" >2.3-2 <span class="Heading" >Character Tables of Groups of the Structure <span class="SimpleMath" >G.S_3</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7D3EF3BC83BE05CF" >2.3-3 <span class="Heading" >Character Tables of Groups of the Structure <span class="SimpleMath" >G.2^2</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X81464C4B8178C85A" >2.3-4 <span class="Heading" >Character Tables of Groups of the Structure <span class="SimpleMath" >2^2.G</span >span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X86CF6A607B0827EE" >2.3-5 <span class="Heading" ><span class="SimpleMath" >p</span >-Modular Tables of Extensions by <span class="SimpleMath" >p</span >-singular Automorphisms</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X788591D78451C024" >2.3-6 <span class="Heading" >Character Tables of Subdirect Products of Index Two (July 2007)</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X817D2134829FA8FA" >2.4 <span class="Heading" >Examples for the Type <span class="SimpleMath" >M.G.A</span ></span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7F2DBAB48437052C" >2.4-1 <span class="Heading" >Character Tables of Dihedral Groups</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7925DBFA7C5986B5" >2.4-2 <span class="Heading" >An <span class="SimpleMath" >M.G.A</span > Type Example with <span class="SimpleMath" >M</span > noncentral in <span class="SimpleMath" >M.G</span > (May 2004)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7ED45AB379093A70" >2.4-3 <span class="Heading" ><strong class="pkg" >Atlas</strong > Tables of the Type <span class="SimpleMath" >M.G.A</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7A236EDE7A7A28F9" >2.4-4 <span class="Heading" >More <strong class="pkg" >Atlas</strong > Tables of the Type <span class="SimpleMath" span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X794EC2FD7F69B4E6" >2.4-5 <span class="Heading" >The Character Tables of <span class="SimpleMath" >4_2.L_3(4).2_3</span > and <span class="SimpleMath" >12_2.L_3(4).2_3</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7E3E748E85AEDDB3" >2.4-6 <span class="Heading" >The Character Tables of <span class="SimpleMath" >12_1.U_4(3).2_2' and
<
span class=
"SimpleMath" >12_2.U_4(3).2_3
' (December 2015)
</
span >
<
span class=
"ContSS" ><
br /><
span class=
"nocss" > </
span ><a href=
"chap2.html#X8379003582D06130" >2
.4-7 <span class="Heading" >Groups of the Structures <span class="SimpleMath" >3.U_3(8).3_1</span > and <span class="SimpleMath" >3.U_3(8).6</span >span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7B46C77B850D3B4D" >2.4-8 <span class="Heading" >The Character Table of <span class="SimpleMath" >(2^2 × F_4(2)):2 < B</span >span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X8254AA4A843F99BE" >2.4-9 <span class="Heading" >The Character Table of <span class="SimpleMath" >2.(S_3 × Fi_22.2) < 2.B</span > (March 2003)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7AF125168239D208" >2.4-10 <span class="Heading" >The Character Table of <span class="SimpleMath" >(2 × 2.Fi_22):2 < Fi_24</span > (November 2008)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X79C93F7D87D9CF1D" >2.4-11 <span class="Heading" >The Character Table of <span class="SimpleMath" >S_3 × 2.U_4(3).2_2 ≤ 2.Fi_22</span > (September 2002)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X83724BCE86FCD77B" >2.4-12 <span class="Heading" >The Character Table of <span class="SimpleMath" >4.HS.2 ≤ HN.2</span > (May 2002)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7E9A88DA7CBF6426" >2.4-13 <span class="Heading" >The Character Tables of <span class="SimpleMath" >4.A_6.2_3</span >, <span "SimpleMath" >12.A_6.2_3</span >,span class="SimpleMath" >4.L_2(25).2_3</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7BD79BA37C3E729B" >2.4-14 <span class="Heading" >The Character Table of <span class="SimpleMath" >4.L_2(49).2_3</span > (December 2020)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X817A961487D2DFD1" >2.4-15 <span class="Heading" >The Character Table of <span class="SimpleMath" >4.L_2(81).2_3</span > (December 2020)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7AF324AF7A54798F" >2.4-16 <span class="Heading" >The Character Table of <span class="SimpleMath" >9.U_3(8).3_3</span > (March 2017)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7E0C603880157C4E" >2.4-17 <span class="Heading" >Pseudo Character Tables of the Type <span class="SimpleMath" >M.G.A</span > (May 2004)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X844185EF7A8F2A99" >2.4-18 <span class="Heading" >Some Extra-ordinary <span class="SimpleMath" >p</span >-Modular Tables of the Type <span class="SimpleMath" >M.G.A</span >span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7F50C782840F06E4" >2.5 <span class="Heading" >Examples for the Type <span class="SimpleMath" >G.S_3</span ></span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7F0DC29F874AA09F" >2.5-1 <span class="Heading" >Small Examples</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X80F9BC057980A9E9" >2.5-2 <span class="Heading" ><strong class="pkg" >Atlas</strong > Tables of the Type <span class="SimpleMath" >G.S_3</span ></span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7EA489E07D7C7D86" >2.6 <span class="Heading" >Examples for the Type <span class="SimpleMath" >G.2^2</span ></span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X8054FDE679053B1C" >2.6-1 <span class="Heading" >The Character Table of <span class="SimpleMath" >A_6.2^2</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7FEC3AB081487AF2" >2.6-2 <span class="Heading" ><strong class="pkg" >Atlas</strong > Tables of the Type <span class="SimpleMath" >G.2^2</span > – Easy Cases</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X869B65D3863EDEC3" >2.6-3 <span class="Heading" >The Character Table of <span class="SimpleMath" >S_4(9).2^2</span > (September 2011)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7B38006380618543" >2.6-4 <span class="Heading" >The Character Tables of Groups of the Type <span class="SimpleMath" >2.L_3(4).2^2</span >span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X79818ABD7E972370" >2.6-5 <span class="Heading" >The Character Tables of Groups of the Type <span class="SimpleMath" >6.L_3(4).2^2</span >span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X878889308653435F" >2.6-6 <span class="Heading" >The Character Tables of Groups of the Type <span class="SimpleMath" >2.U_4(3).2^2</span >span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7DC42AE57E9EED4D" >2.6-7 <span class="Heading" >The Character Tables of Groups of the Type <span class="SimpleMath" >4_1.L_3(4).2^2</span >span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7E9AF180869B4786" >2.6-8 <span class="Heading" >The Character Tables of Groups of the Type <span class="SimpleMath" >4_2.L_3(4).2^2</span >span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7EAF9CD07E536120" >2.6-9 <span class="Heading" >The Character Table of Aut<span class="SimpleMath" >(L_2(81))</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X78AED04685EDCC19" >2.6-10 <span class="Heading" >The Character Table of <span class="SimpleMath" >O_8^+(3).2^2_111</span ></span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X845BAA2A7FD768B0" >2.7 <span class="Heading" >Examples for the Type <span class="SimpleMath" >2^2.G</span ></span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X87EEBDB987249117" >2.7-1 <span class="Heading" >The Character Table of <span class="SimpleMath" >2^2.Sz(8)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X83652A0282A64D14" >2.7-2 <span class="Heading" ><strong class="pkg" >Atlas</strong > Tables of the Type <span class="SimpleMath" >2^2.G</span > (September 2005)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7F63DDF77870F967" >2.7-3 <span class="Heading" >The Character Table of <span class="SimpleMath" >2^2.O_8^+(3)</span > (March 2009)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X86A1607787DE6BB9" >2.7-4 <span class="Heading" >The Character Table of the Schur Cover of <span class="SimpleMath" >L_3(4)</span >span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X8711DBB083655A25" >2.8 <span class="Heading" >Examples of Extensions by <span class="SimpleMath" >p</span >-singular Automorphisms</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X81C08739850E4AAE" >2.8-1 <span class="Heading" >Some <span class="SimpleMath" >p</span >-Modular Tables of Groups of the Type <span class="SimpleMath" >M.G.A</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7FED618F83ACB7C2" >2.8-2 <span class="Heading" >Some <span class="SimpleMath" >p</span >-Modular Tables of Groups of the Type <span class="SimpleMath" >G.S_3</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7EEF6A7F8683177A" >2.8-3 <span class="Heading" ><span class="SimpleMath" >2</span >-Modular Tables of Groups of the Type <span class="SimpleMath" >G.2^2</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X875F8DD77C0997FA" >2.8-4 <span class="Heading" >The <span class="SimpleMath" >3</span >-Modular Table of <span class="SimpleMath" >U_3(8).3^2</span ></span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7A4D6044865E516B" >2.9 <span class="Heading" >Examples of Subdirect Products of Index Two</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X850FF694801700CF" >2.9-1 <span class="Heading" >Certain Dihedral Groups as Subdirect Products of Index Two</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X80C5D6FA83D7E2CF" >2.9-2 <span class="Heading" >The Character Table of <span class="SimpleMath" >(D_10 × HN).2 < M</span > (June 2008)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X85EECFD47EC252A2" >2.9-3 <span class="Heading" >A Counterexample (August 2015)</span ></a>span >div ></div >div >div class="ContChap" ><a href="chap3.html#X7A80D5ED7D6E57B7" >3 <span class="Heading" >Constructing Character Tables of Central Extensions in <strong class="pkg" >GAP</strong ></span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X87B17873861E2F64" >3.1 <span class="Heading" >Coprime Central Extensions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X85CB2671851D1206" >3.1-1 <span class="Heading" >The Character Table Head </span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7D8F6E5D7D632046" >3.1-2 <span class="Heading" >The Irreducible Characters</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X867D16E07D36560F" >3.1-3 <span class="Heading" >Ordering of Conjugacy Classes</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X813B9F5180A45077" >3.1-4 <span class="Heading" >Compatibility with Smaller Factor Groups</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7A489A5D79DA9E5C" >3.2 <span class="Heading" >Examples</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X861B5C3F7B1F6AB7" >3.2-1 <span class="Heading" >Central Extensions of Simple <strong class="pkg" >Atlas</strong > Groups</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X799ADD5487613BA2" >3.2-2 <span class="Heading" >Central Extensions of Other <strong class="pkg" >Atlas</strong > Groups</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X861F558380FE4812" >3.2-3 <span class="Heading" >Compatible Central Extensions of Maximal Subgroups</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7C73944579D6EE73" >3.2-4 <span class="Heading" >The <code class="code" >2B</code > Centralizer in <span class="SimpleMath" >3.Fi_24' (January 2004)span >div ></div >div >div class="ContChap" ><a href="chap4.html#X7D5919C182B1A462" >4 <span class="Heading" ><strong class="pkg" >GAP</strong > Computations Concerning Hamiltonian Cycles in the Generating Graphs of Finite Groups</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X8389AD927B74BA4A" >4.1 <span class="Heading" >Overview</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7B6AEBDF7B857E2E" >4.2 <span class="Heading" >Theoretical Background</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7AD3962D7AE4ADFB" >4.2-1 <span class="Heading" >Character-Theoretic Lower Bounds for Vertex Degrees</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X825776BA8687E475" >4.2-2 <span class="Heading" >Checking the Criteria</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7B56BE5384BAD54E" >4.3 <span class="Heading" ><strong class="pkg" >GAP</strong > Functions for the Computations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X802B2ED2802334B0" >4.3-1 <span class="Heading" >Computing Vertex Degrees from the Group</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X87FE2DDD7F086D2F" >4.3-2 <span class="Heading" >Computing Lower Bounds for Vertex Degrees</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X8677A8B1788ACD2C" >4.3-3 <span class="Heading" >Evaluating the (Lower Bounds for the) Vertex Degrees</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7A221012861440E2" >4.4 <span class="Heading" >Character-Theoretic Computations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X86CE51E180A3D4ED" >4.4-1 <span class="Heading" >Sporadic Simple Groups</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X867D338F7F453092" >4.4-2 <span class="Heading" >The Monster</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7DC6DFCC83502CC3" >4.4-3 <span class="Heading" >Nonsimple Automorphism Groups of Sporadic Simple Groups</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X8130C9CB7A33140F" >4.4-4 <span class="Heading" >Alternating and Symmetric Groups <span class="SimpleMath" >A_n</span >, <span class="SimpleMath" >S_n</span >,span class="SimpleMath" >5 ≤ n ≤ 13</span ></span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X83DACCF07EF62FAE" >4.5 <span class="Heading" >Computations With Groups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7B9ADC91802EE09F" >4.5-1 <span class="Heading" >Nonabelian Simple Groups of Order up to <span class="SimpleMath" >10^7</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X8033892B7FD6E62B" >4.5-2 <span class="Heading" >Nonsimple Groups with Nonsolvable Socle of Order at most <span class="SimpleMath" span ></span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X84E62545802FAB30" >4.6 <span class="Heading" >The Groups <span class="SimpleMath" >PSL(2,q)</span ></span ></a>span >div >div >div class="ContChap" ><a href="chap5.html#X8703EFEE81DDE3DD" >5 <span class="Heading" ><strong class="pkg" >GAP</strong > Computations with <span class="SimpleMath" >O_8^+(5).S_3</span > and <span class="SimpleMath" >O_8^+(2).S_3</span ></span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X8389AD927B74BA4A" >5.1 <span class="Heading" >Overview</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X85FF559084C08F0F" >5.2 <span class="Heading" >Constructing Representations of <span class="SimpleMath" >M.2</span > and <span class="SimpleMath" >S.2</span ></span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X7FEE53AB845B9327" >5.2-1 <span class="Heading" >A Matrix Representation of the Weyl Group of Type <span class="SimpleMath" >E_8</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X7C8AA7747F160F8A" >5.2-2 <span class="Heading" >Embedding the Weyl group of Type <span class="SimpleMath" >E_8</span > into GO<span class="SimpleMath" >^+(8,5)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X83E3E79F8724C365" >5.2-3 <span class="Heading" >Compatible Generators of <span class="SimpleMath" >M</span >, <span class="SimpleMath" >M.2</span >, <span class="SimpleMath" >S</span >, and <span class="SimpleMath" >S.2</span ></span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X83F897DD7C48511C" >5.3 <span class="Heading" >Constructing Representations of <span class="SimpleMath" >M.3</span > and <span class="SimpleMath" >S.3</span ></span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X7B7561D0855EC4F1" >5.3-1 <span class="Heading" >The Action of <span class="SimpleMath" >M.3</span > on <span class="SimpleMath" >M</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X8246803779EB8FEE" >5.3-2 <span class="Heading" >The Action of <span class="SimpleMath" >S.3</span > on <span class="SimpleMath" >S</span ></span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X816AFA187E95C018" >5.4 <span class="Heading" >Constructing Compatible Generators of <span class="SimpleMath" >H</span > and <span class="SimpleMath" >G</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X83F0387D789709D1" >5.5 <span class="Heading" >Application: Regular Orbits of <span class="SimpleMath" >H</span > on <span class="SimpleMath" >G/H</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X7F0C266082BE1578" >5.6 <span class="Heading" >Appendix: The Permutation Character <span class="SimpleMath" >(1_H^G)_H</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X7F3A630780F8E262" >5.7 <span class="Heading" >Appendix: The Data File</span ></a>span >div >div >div class="ContChap" ><a href="chap6.html#X7EF73AA88384B5F3" >6 <span class="Heading" >Solvable Subgroups of Maximal Order in Sporadic Simple Groups</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X7F817DC57A69CF0D" >6.1 <span class="Heading" >The Result</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X876F77197B2FB84A" >6.2 <span class="Heading" >The Approach</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X792957AB7B24C5E0" >6.2-1 <span class="Heading" >Use the Table of Marks</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7B39A4467A1CCF8A" >6.2-2 <span class="Heading" >Use Information from the Character Table Library</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X834298A87BF43AAF" >6.3 <span class="Heading" >Cases where the Table of Marks is available in <strong class="pkg" >GAP</strong ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X85559C0F7AA73E48" >6.4 <span class="Heading" >Cases where the Table of Marks is not available in <strong class="pkg" >GAP</strong ></span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7E393459822E78B5" >6.4-1 <span class="Heading" ><span class="SimpleMath" >G = Ru</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7AFF09337CCB7745" >6.4-2 <span class="Heading" ><span class="SimpleMath" >G = Suz</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7969AE067D3862A3" >6.4-3 <span class="Heading" ><span class="SimpleMath" >G = ON</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X84921B85845EDA31" >6.4-4 <span class="Heading" ><span class="SimpleMath" >G = Co_2</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7D777A0D82BE8498" >6.4-5 <span class="Heading" ><span class="SimpleMath" >G = Fi_22</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7D9DB76A861A6F62" >6.4-6 <span class="Heading" ><span class="SimpleMath" >G = HN</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X83E6436678AF562C" >6.4-7 <span class="Heading" ><span class="SimpleMath" >G = Ly</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7D6CF8EC812EF6FB" >6.4-8 <span class="Heading" ><span class="SimpleMath" >G = Th </span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7A07090483C935DC" >6.4-9 <span class="Heading" ><span class="SimpleMath" >G = Fi_23</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7D028E9E7CB62A4F" >6.4-10 <span class="Heading" ><span class="SimpleMath" >G = Co_1</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X84208AB781344A9D" >6.4-11 <span class="Heading" ><span class="SimpleMath" >G = J_4</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7BC589718203F125" >6.4-12 <span class="Heading" ><span class="SimpleMath" >G = Fi_24^' span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7EDF990985573EB6" >6.4-13 <span class="Heading" ><span class="SimpleMath" >G = B</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X87D468D07D7237CB" >6.4-14 <span class="Heading" ><span class="SimpleMath" >G = M</span ></span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X7CD8E04C7F32AD56" >6.5 <span class="Heading" >Proof of the Corollary</span ></a>span >div >div >div class="ContChap" ><a href="chap7.html#X8102827B85FE3BCA" >7 <span class="Heading" >Large Nilpotent Subgroups of Sporadic Simple Groups</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X7F817DC57A69CF0D" >7.1 <span class="Heading" >The Result</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X787B841383A16711" >7.2 <span class="Heading" >The Proof</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X798EACC07F6C36D9" >7.3 <span class="Heading" >Alternative: Use <strong class="pkg" >GAP</strong >'s Tables of Marks span >div >div >div class="ContChap" ><a href="chap8.html#X7A7EEBE9858333E1" >8 <span class="Heading" >Permutation Characters in <strong class="pkg" >GAP</strong ></span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X86A1325B82E5AECD" >8.1 <span class="Heading" >Some Computations with <span class="SimpleMath" >M_24</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X79C9051F805851DB" >8.2 <span class="Heading" >All Possible Permutation Characters of <span class="SimpleMath" >M_11</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X81A5FC968782CFC3" >8.3 <span class="Heading" >The Action of <span class="SimpleMath" >U_6(2)</span > on the Cosets of <span class="SimpleMath" >M_22</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X7EE1811C8496C428" >8.4 <span class="Heading" >Degree <span class="SimpleMath" >20736</span > Permutation Characters of <span class="SimpleMath" >U_6(2)</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X7DC6A6E785A347C8" >8.5 <span class="Heading" >Degree <span class="SimpleMath" >57572775</span > Permutation Characters of <span class="SimpleMath" >O_8^+(3)</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X792D2C2380591D8D" >8.6 <span class="Heading" >The Action of <span class="SimpleMath" >O_7(3).2</span > on the Cosets of <span class="SimpleMath" >2^7.S_7</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X875B361C8512939F" >8.7 <span class="Heading" >The Action of <span class="SimpleMath" >O_8^+(3).2_1</span > on the Cosets of <span class="SimpleMath" >2^7.A_8</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X7B1DFAF98182CFF4" >8.8 <span class="Heading" >The Action of <span class="SimpleMath" >S_4(4).4</span > on the Cosets of <span class="SimpleMath" >5^2.[2^5]</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X7F04F0C684AA8B30" >8.9 <span class="Heading" >The Action of <span class="SimpleMath" >Co_1</span > on the Cosets of Involution Centralizers</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X8230719D8538384B" >8.10 <span class="Heading" >The Multiplicity Free Permutation Characters of <span "SimpleMath" >G_2(3)</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X7E3E326C7CB0E2CD" >8.11 <span class="Heading" >Degree <span class="SimpleMath" >11200</span > Permutation Characters of <span class="SimpleMath" >O_8^+(2)</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X7D8572E68194CBB9" >8.12 <span class="Heading" >A Proof of Nonexistence of a Certain Subgroup</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X8068E9DA7CD03BF2" >8.13 <span class="Heading" >A Permutation Character of the Lyons group</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X87D6C1A67CC7EE0A" >8.14 <span class="Heading" >Identifying two subgroups of Aut<span class="SimpleMath" >(U_3(5))</span > (October 2001)</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X793669787CF73A55" >8.15 <span class="Heading" >A Permutation Character of Aut<span class="SimpleMath" >(O_8^+(2))</span > (October 2001)</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X8337F3C682B6BE63" >8.16 <span class="Heading" >Four Primitive Permutation Characters of the Monster Group</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X78A8A1248336DD26" >8.16-1 <span class="Heading" >The Subgroup <span class="SimpleMath" >2^2.2^11.2^22.(S_3 × M_24)</span span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X79E9247182B20474" >8.16-2 <span class="Heading" >The Subgroup <span class="SimpleMath" >2^3.2^6.2^12.2^18.(L_3(2) × 3.S_6)</span >span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7BC36C597E542DEE" >8.16-3 <span class="Heading" >The Subgroup <span class="SimpleMath" >2^5.2^10.2^20.(S_3 × L_5(2))</span >span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7F2ABD3E7AFF5F6E" >8.16-4 <span class="Heading" >The Subgroup <span class="SimpleMath" >2^{10+16}.O_10^+(2)</span > (November 2009)</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X87D11B097D95D027" >8.17 <span class="Heading" >A permutation character of the Baby Monster (June 2012)</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X86827FA97D27F3A2" >8.18 <span class="Heading" >A permutation character of <span class="SimpleMath" >2.B</span > (October 2017)</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X849F0EA6807C9B19" >8.19 <span class="Heading" >Generation of sporadic simple groups by <span class="SimpleMath" >π</span >- and <span class="SimpleMath" >π' -subgroups (December 2021)span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X839272078243F4DB" >8.19-1 <span class="Heading" >Special Arguments for the Monster Group that are no longer needed</span ></a>span >div ></div >div >div class="ContChap" ><a href="chap9.html#X7A03A83E87FB1189" >9 <span class="Heading" >Ambiguous Class Fusions in the <strong class="pkg" >GAP</strong > Character Table Library</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X784492877DB04FE9" >9.1 <span class="Heading" >Some <strong class="pkg" >GAP</strong > Utilities</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X7EA839057D3AD3B4" >9.2 <span class="Heading" >Fusions Determined by Factorization through Intermediate Subgroups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X78DCEEFD85FF1EE2" >9.2-1 <span class="Heading" ><span class="SimpleMath" >Co_3N5 → Co_3</span > (September 2002)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X86BCEA907EC4C833" >9.2-2 <span class="Heading" ><span class="SimpleMath" >31:15 → B</span > (March 2003)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7C719F527831F35A" >9.2-3 <span class="Heading" ><span class="SimpleMath" >SuzN3 → Suz</span > (September 2002)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X828879F481EF30DD" >9.2-4 <span class="Heading" ><span class="SimpleMath" >F_{3+}N5 → F_{3+}</span > (March 2002)</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X7981579278F81AC6" >9.3 <span class="Heading" >Fusions Determined Using Commutative Diagrams Involving Smallerspan ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7F5186E28201B027" >9.3-1 <span class="Heading" ><span class="SimpleMath" >BN7 → B</span > (March 2002)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X79710B137B5BB1B8" >9.3-2 <span class="Heading" ><span class="SimpleMath" >(A_4 × O_8^+(2).3).2 → Fi_24^' (November 2002)span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X85822C647B29117B" >9.3-3 <span class="Heading" ><span class="SimpleMath" >A_6 × L_2(8).3 → Fi_24^' (November 2002)span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X81A607758682D9A9" >9.3-4 <span class="Heading" ><span class="SimpleMath" >(3^2:D_8 × U_4(3).2^2).2 → B</span > (June 2007)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7962DD4387D63675" >9.3-5 <span class="Heading" ><span class="SimpleMath" >7^1+4:(3 × 2.S_7) → M</span > (May 2009)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X860B6C30812DE3FC" >9.3-6 <span class="Heading" ><span class="SimpleMath" >3^7.O_7(3):2 → Fi_24</span > (November 2010)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7C3AC42F8342EE2E" >9.3-7 <span class="Heading" ><span class="SimpleMath" >^2E_6(2)N3C → ^2E_6(2)</span > (January 2019)</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X84F966E2824F5D52" >9.4 <span class="Heading" >Fusions Determined Using Commutative Diagrams Involving Factorspan ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7F2B104686509CAA" >9.4-1 <span class="Heading" ><span class="SimpleMath" >3.A_7 → 3.Suz</span > (December 2010)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X82FB71647D37F4FD" >9.4-2 <span class="Heading" ><span class="SimpleMath" >S_6 → U_4(2)</span > (September 2011)</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X7CFBC41B818A318C" >9.5 <span class="Heading" >Fusions Determined Using Commutative Diagrams Involvingspan ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7E91F8707BA93081" >9.5-1 <span class="Heading" ><span class="SimpleMath" >U_3(8).3_1 → ^2E_6(2)</span > (December 2010)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X81B37EF378E89E00" >9.5-2 <span class="Heading" ><span class="SimpleMath" >L_3(4).2_1 → U_6(2)</span > (December 2010)</span span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X85E2A6F480026C95" >9.6 <span class="Heading" >Conditions Imposed by Brauer Tables</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7ACC7F588213D5D5" >9.6-1 <span class="Heading" ><span class="SimpleMath" >L_2(16).4 → J_3.2</span > (January 2004)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7ACB86CB82ED49D1" >9.6-2 <span class="Heading" ><span class="SimpleMath" >L_2(17) → S_8(2)</span > (July 2004)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7DED4C437D479226" >9.6-3 <span class="Heading" ><span class="SimpleMath" >L_2(19) → J_3</span > (April 2003)</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X8225D9FA80A7D20F" >9.7 <span class="Heading" >Fusions Determined by Information about the Groups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7AE2962E82B4C814" >9.7-1 <span class="Heading" ><span class="SimpleMath" >U_3(3).2 → Fi_24^' (November 2002)span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X83061094871EE241" >9.7-2 <span class="Heading" ><span class="SimpleMath" >L_2(13).2 → Fi_24^' (September 2002)span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7E9C203C7C4D709D" >9.7-3 <span class="Heading" ><span class="SimpleMath" >M_11 → B</span > (April 2009)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X85821D748716DC7E" >9.7-4 <span class="Heading" ><span class="SimpleMath" >L_2(11):2 → B</span > (April 2009)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X828D81487F57D612" >9.7-5 <span class="Heading" ><span class="SimpleMath" >L_3(3) → B</span > (April 2009)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7B4E13337D66020F" >9.7-6 <span class="Heading" ><span class="SimpleMath" >L_2(17).2 → B</span > (March 2004)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X8528432A84851F7B" >9.7-7 <span class="Heading" ><span class="SimpleMath" >L_2(49).2_3 → B</span > (June 2006)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7EAD52AA7A28D956" >9.7-8 <span class="Heading" ><span class="SimpleMath" >2^3.L_3(2) → G_2(5)</span > (January 2004)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X79617107849A6CEA" >9.7-9 <span class="Heading" ><span class="SimpleMath" >5^{1+4}.2^{1+4}.A_5.4 → B</span > (April 2009)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X85C48EEB7B711C09" >9.7-10 <span class="Heading" >The fusion from the character table of <span class="SimpleMath" >7^2:2L_2(7).2</span >table of marks (January 2004)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7B1C689C7EFD07CB" >9.7-11 <span class="Heading" ><span class="SimpleMath" >3 × U_4(2) → 3_1.U_4(3)</span > (March 2010)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7A94F78C792122D5" >9.7-12 <span class="Heading" ><span class="SimpleMath" >2.3^4.2^3.S_4 → 2.A12</span > (September 2011)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7E2AF30C7E8F89F9" >9.7-13 <span class="Heading" ><span class="SimpleMath" >127:7 → L_7(2)</span > (January 2012)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X80051B297DF244CF" >9.7-14 <span class="Heading" ><span class="SimpleMath" >L_2(59) → M</span > (May 2009) – Do not use this!</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X8409DA2E83A41ABE" >9.7-15 <span class="Heading" ><span class="SimpleMath" >L_2(71) → M</span > (May 2009)</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X78B3B1BE7A2CA4D1" >9.7-16 <span class="Heading" ><span class="SimpleMath" >L_2(41) → M</span > (April 2012)</span ></a>span >div ></div >div >div class="ContChap" ><a href="chap10.html#X831E9D0A7A2DBC72" >10 <span class="Heading" ><strong class="pkg" >GAP</strong > computations needed in the proof of"chapBib.html#biBDNT" >[DNT13, Theorem 6.1 (ii)]</a></span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X82BDD020860C6E95" >10.1 <span class="Heading" ><span class="SimpleMath" >G/N ≅ Sz(8)</span > and <span class="SimpleMath" >|N| = 2^12</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X7C01350E8217B0B1" >10.2 <span class="Heading" ><span class="SimpleMath" >G/N ≅ M_22</span > and <span class="SimpleMath" >|N| = 2^10</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X7E356703856DF22E" >10.3 <span class="Heading" ><span class="SimpleMath" >G/N ≅ J_2</span > and <span class="SimpleMath" >|N| = 2^12</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X797E2EDB78F05F6E" >10.4 <span class="Heading" ><span class="SimpleMath" >G/N ≅ J_2</span > and <span class="SimpleMath" >|N| = 5^14</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X828AECAE82B0CEB6" >10.5 <span class="Heading" ><span class="SimpleMath" >G/N ≅ J_2</span > and <span class="SimpleMath" >|N| = 2^28</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X81AB173981E3EED7" >10.6 <span class="Heading" ><span class="SimpleMath" >G/N ≅ ^3D_4(2)</span > and <span "SimpleMath" >|N| = 2^26</span ></span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X83B044547B96B7A5" >10.7 <span class="Heading" ><span class="SimpleMath" >G/N ≅ ^3D_4(2)</span > and <span "SimpleMath" >|N| = 3^25</span ></span ></a>span >div >div >div class="ContChap" ><a href="chap11.html#X7BE9906583D0FCEC" >11 <span class="Heading" ><strong class="pkg" >GAP</strong > Computations Concerning Probabilistic Generation of Finitespan ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11.html#X8389AD927B74BA4A" >11.1 <span class="Heading" >Overview</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11.html#X7B4649CF7B7CFAA1" >11.2 <span class="Heading" >Prerequisites</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7B6AEBDF7B857E2E" >11.2-1 <span class="Heading" >Theoretical Background</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X79D7312484E78274" >11.2-2 <span class="Heading" >Computational Criteria</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11.html#X7B56BE5384BAD54E" >11.3 <span class="Heading" ><strong class="pkg" >GAP</strong > Functions for the Computations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X806328747D1D4ECC" >11.3-1 <span class="Heading" >General Utilities</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7A221012861440E2" >11.3-2 <span class="Heading" >Character-Theoretic Computations</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X83DACCF07EF62FAE" >11.3-3 <span class="Heading" >Computations with Groups</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11.html#X7A221012861440E2" >11.4 <span class="Heading" >Character-Theoretic Computations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X86CE51E180A3D4ED" >11.4-1 <span class="Heading" >Sporadic Simple Groups</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X821778BC7D665AB4" >11.4-2 <span class="Heading" >No longer necessary computations for the Baby Monster and the Monster</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X84E9D10F80A74A53" >11.4-3 <span class="Heading" >Automorphism Groups of Sporadic Simple Groups</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X80DA58F187CDCF5F" >11.4-4 <span class="Heading" >Other Simple Groups – Easy Cases</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7B1E26D586337487" >11.4-5 <span class="Heading" >Automorphism Groups of other Simple Groups – Easy Cases</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X78B856907ED13545" >11.4-6 <span class="Heading" ><span class="SimpleMath" >O_8^-(3)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X84AB334886DCA746" >11.4-7 <span class="Heading" ><span class="SimpleMath" >O_10^+(2)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X84E3E4837BB93977" >11.4-8 <span class="Heading" ><span class="SimpleMath" >O_10^-(2)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X8307367E7C7C3BCE" >11.4-9 <span class="Heading" ><span class="SimpleMath" >O_12^+(2)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X834FE1B58119A5FF" >11.4-10 <span class="Heading" ><span class="SimpleMath" >O_12^-(2)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7C5980A385C088FA" >11.4-11 <span class="Heading" ><span class="SimpleMath" >S_6(4)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X829EDF7F7C0BCB8E" >11.4-12 <span class="Heading" ><span class="SimpleMath" >∗</span > <span class="SimpleMath" >S_6(5)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X85162B297E4B67EB" >11.4-13 <span class="Heading" ><span class="SimpleMath" >S_8(3)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X8495C2BF7B6EFFEF" >11.4-14 <span class="Heading" ><span class="SimpleMath" >U_4(4)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7A3BB5AA83A2BDF3" >11.4-15 <span class="Heading" ><span class="SimpleMath" >U_6(2)</span ></span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11.html#X8237B8617D6F6027" >11.5 <span class="Heading" >Computations using Groups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X815320787B601000" >11.5-1 <span class="Heading" ><span class="SimpleMath" >A_2m+1</span >, <span class="SimpleMath" >2 ≤ m ≤ 11</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7B5321337B28100B" >11.5-2 <span class="Heading" ><span class="SimpleMath" >A_5</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X82C3B4287B0C7BEE" >11.5-3 <span class="Heading" ><span class="SimpleMath" >A_6</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X85B3C7217B105D4D" >11.5-4 <span class="Heading" ><span class="SimpleMath" >A_7</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X84EA645A82E2BAFB" >11.5-5 <span class="Heading" ><span class="SimpleMath" >L_d(q)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X855460BE787188B9" >11.5-6 <span class="Heading" ><span class="SimpleMath" >∗</span > <span class="SimpleMath" >L_d(q)</span span class="SimpleMath" >d</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7EA88CEF81962F3F" >11.5-7 <span class="Heading" >Automorphic Extensions of <span class="SimpleMath" >L_d(q)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7C8806DB8588BB51" >11.5-8 <span class="Heading" ><span class="SimpleMath" >L_3(2)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7B7061917ED3714D" >11.5-9 <span class="Heading" ><span class="SimpleMath" >M_11</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X82E0F48A7FF82BB3" >11.5-10 <span class="Heading" ><span class="SimpleMath" >M_12</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7FF2E8F27FBEB65C" >11.5-11 <span class="Heading" ><span class="SimpleMath" >O_7(3)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7F80F2527C424AA4" >11.5-12 <span class="Heading" ><span class="SimpleMath" >O_8^+(2)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X78F0815B86253A1F" >11.5-13 <span class="Heading" ><span class="SimpleMath" >O_8^+(3)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X85BACC4A83F73392" >11.5-14 <span class="Heading" ><span class="SimpleMath" >O^+_8(4)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X86EC26F78609618E" >11.5-15 <span class="Heading" ><span class="SimpleMath" >∗</span > <span class="SimpleMath" >O_9(3)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X8393978A8773997E" >11.5-16 <span class="Heading" ><span class="SimpleMath" >O_10^-(3)</span ></span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7BBBEEEF834F1002" >11.5-17 <span class="Heading" ><span class="SimpleMath" >O_14^-(2)</span ></span ></a>span >quality 99%
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