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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> <html> <head> <meta http-equiv="content-type" content="text/html; charset=ISO-8859-1"> <title>AboutHap</title> </head> <body style="color: rgb(0, 0, 153); background-color: rgb(204, 255, 255);" link="#000066" vlink="#000066" alink="#000066"> <br> <table style="text-align: left; margin-left: auto; margin-right: auto; color: rgb(0, 0, 102);" border="0" cellpadding="20" cellspacing="10"> <tbody> <tr align="center"> <th style="vertical-align: top;"> <table style="width: 100%; text-align: left;" cellpadding="2" cellspacing="2"> <tbody> <tr> <td style="vertical-align: top;"><a href="aboutCoefficientSequence.html"><small style="color: rgb(0, 0, 102);">Previous</small></a><br> </td> <td style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big><span style="font-weight: bold;">About HAP: Third homology of some superperfect groups<br> </span></big></td> <td style="text-align: right; vertical-align: top;"><a href="aboutPeriodic.html"><small style="color: rgb(0, 0, 102);">next</small></a><br> </td> </tr> </tbody> </table> <big><span style="font-weight: bold;"></span></big><br> </th> </tr> <tr> <td style="vertical-align: top; background-color: rgb(255, 255, 255);"> <div style="text-align: left;">A group G is said to be <span style="font-style: italic;">superperfect</span> if it is perfect and its second integral homology is trivial. Using the library of perfect groups compiled by Holt and Pleskin, we (<span style="color: rgb(255, 0, 0);">very partially</span>) list below the superperfect groups of order at most 68880 together with their third integral homology. Further information on these superperfect groups can be obtained using the GAP command<br> <div style="text-align: center;"><br> <span style="font-family: helvetica,arial,sans-serif;">DisplayInformationPerfectGrou <br> </div> </div> <br> where "order" is the order of the group and "number" is the number of the group given in the table. <br> <br> <span style="color: rgb(255, 0, 0);">(Work on the table is to be continued ... )</span><br> </td> </tr> <tr> <td style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;"> <table style="margin-left: auto; margin-right: auto; text-align: left; width: 555px; height: 823 border="0" cellpadding="2" cellspacing="2"> <tbody> <tr> <td style="vertical-align: top;"><br> </td> <td style="vertical-align: top; text-align: center;"><big><big><span style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;"><small style="text-decoration: underline;">List of superperfect groups</small><br style="text-decoration: underline;"> <br> <br> </span></big></big> </td> <td style="vertical-align: top;"><br> </td> </tr> <tr> <td style="vertical-align: top; font-family: helvetica,arial,sans-serif; text-align: left style="color: rgb(0, 0, 0);">Order </span><br> </big></td> <td style="vertical-align: top; text-align: left;"><big><span style="font-family: helvetica,arial,sans-serif; color: rgb(0, 0, 0);">Perfect Group Number</span><br> </big></td> <td style="vertical-align: top; text-align: left;"><big><span style="font-family: helvetica,arial,sans-serif; color: rgb(0, 0, 0);">Third integral homology </span><br> </big></td> </tr> <tr> <td style="vertical-align: top; text-align: left;"><br> </td> <td style="vertical-align: top; text-align: left;"><br> </td> <td style="vertical-align: top; text-align: left;"><br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left;"><span style="color: rgb(0, 0, 0);">120</span><br> </td> <td style="vertical-align: top; text-align: left;"><span style="color: rgb(0, 0, 0);">1</span><br> </td> <td style="vertical-align: top; text-align: left;"><span style="color: rgb(0, 0, 0);">[ 8, 3, 5 ]</span><br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">336<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[16,3]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">504<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 2, 9, 7 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1320<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 8, 3, 5 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">2160<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 16, 3, 3, 5 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">2184<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 8, 3, 7 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">2688<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">3<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 8, 8, 3 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">3840<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">6<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">???<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">3840<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">7<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">???<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">4080<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 3, 5, 17 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">4896<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 32, 9 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">5376<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">???<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">5616<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 8, 3 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">6048<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 8, 9, 5 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">7920<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 8 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">9720<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">2<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 8, 9, 5 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">12144<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 16, 3, 11 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">14400<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 8, 8, 3, 3, 5, 5 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">15000<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">2<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 8, 3, 5, 25 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">15000<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">3<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">???<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">15120<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 16, 3, 3 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">15360<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">4,6-7: Not yet checked if these are superperfect<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);"><br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">15600<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 16, 3, 13 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">19656<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 8, 3, 7, 13 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">21504<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">5-22: Not yet checked if these are superferfect<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);"><br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">24360<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 8, 3, 5, 7 ]<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">29160<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">???<br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">29160<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">3: Not yet checked if this is superperfect<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);"><br> </td> </tr> <tr> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">29760<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; text-align: left; color: rgb(0, 0, 0);">[ 64, 3, 5 ]<br> </td> </tr> <tr> <td style="vertical-align: top;"><span style="color: rgb(0, 0, 0);">32256</span><br> </td> <td style="vertical-align: top;"><span style="color: rgb(0, 0, 0);">1-2: Not yet checked if these are superperfect</span><br> </td> <td style="vertical-align: top;"><br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">32736<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">[ 3, 11, 31 ]<br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">34560<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">4<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">???<br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">37500<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">1: Not yet checked if this is superperfect<br> </td> <td style="vertical-align: top;"><br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">40320<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">[ 8, 16, 3, 3, 5 ]<br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">40320<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">3<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">???<br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">43008<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">1-25: Not yet checked if these are superperfect<br> </td> <td style="vertical-align: top;"><br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">46080<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">1: Not yet checked if this is superperfect<br> </td> <td style="vertical-align: top;"><br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">50616<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">[ 8, 9, 19 ]<br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">51840<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">???<br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">57624<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">1-2: Not yet checked if these are superperfect<br> </td> <td style="vertical-align: top;"><br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">58320<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">2: Not yet checked if this is superperfect<br> </td> <td style="vertical-align: top;"><br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">62400<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">???<br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">64512<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">1-4: Not yet checked if these are superperfect<br> </td> <td style="vertical-align: top;"><br> </td> </tr> <tr> <td style="vertical-align: top; color: rgb(0, 0, 0);">68880<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">1<br> </td> <td style="vertical-align: top; color: rgb(0, 0, 0);">[ 16, 3, 5, 7 ]<br> </td> </tr> </tbody> </table> <br> </td> </tr> <tr> <td style="vertical-align: top;"> <table style="margin-left: auto; margin-right: auto; width: 100%; text-align: left;" border="0" cellpadding="2" cellspacing="2"> <tbody> <tr> <td style="vertical-align: top;"><a style="color: rgb(0, 0, 102);" href="aboutCoefficientSequence.html">Previous Page</a><br> </td> <td style="text-align: center; vertical-align: top;"><a href="aboutContents.html"><span style="color: rgb(0, 0, 102);">Contents</span></a><br> </td> <td style="text-align: right; vertical-align: top;"><a href="aboutPeriodic.html"><span style="color: rgb(0, 0, 102);">Next page</span><br> </a> </td> </tr> </tbody> </table> <a href="aboutTopology.html"><br> </a> </td> </tr> </tbody> </table> <br> <br> </body> </html>
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2026-11-04