Symmetric Group 8<title> </head> <body> <big style="font-weight: bold;"><big>Group: S8</big></big><span style="font-weight: bold;"><br> Order: 2<sup>7</sup>·3<sup>2</sup>·5·7 <br><br> <table style="text-align: center; vertical-align: middle; width: 100%;" border="1" cellpadding="2" cellspacing="2"> <tbody> <tr> <td>Characteristic</td> <td>Block</td> <td>Dimension</td> <td>Cartan Matrix</td> <td>Generators</td> <td>Links</td> </tr> <tr> <td colspan="1" rowspan="2">2</td> <td>1</td> <td>4496</td><td><table style="text-align: left; margin-left: auto; margin-right: auto;"> <tbody> <tr> <td style="text-align: center;">6</td> <td style="text-align: center;">12</td> <td style="text-align: center;">4</td> <td style="text-align: center;">6</td> <td style="text-align: center;">3</td> </tr> <tr> <td style="text-align: center;">12</td> <td style="text-align: center;">32</td> <td style="text-align: center;">16</td> <td style="text-align: center;">16</td> <td style="text-align: center;">8</td> </tr> <tr> <td style="text-align: center;">4</td> <td style="text-align: center;">16</td> <td style="text-align: center;">24</td> <td style="text-align: center;">16</td> <td style="text-align: center;">12</td> </tr> <tr> <td style="text-align: center;">6</td> <td style="text-align: center;">16</td> <td style="text-align: center;">16</td> <td style="text-align: center;">16</td> <td style="text-align: center;">8</td> </tr> <tr> <td style="text-align: center;">3</td> <td style="text-align: center;">8</td> <td style="text-align: center;">12</td> <td style="text-align: center;">8</td> <td style="text-align: center;">9</td> </tr> </tbody> </table> </td> <td>d, bcba, a, <br>z<sub>8</sub>, z<sub>4</sub></td> <td style="font-weight: bold;"><a style="font-weight: normal;" href="S8files/S8mod2b1.html">Condensation Information</a><br> <a style="font-weight: normal;" href="S8files/S8mod2b1extdim.html">Ext Dimensions</a><br> <a style="font-weight: normal;" href="S8files/S8mod2b1.gap.gz">Basic Algebra</a></td> </tr> <tr> <td>2</td> <td>108</td><td><table style="text-align: left; margin-left: auto; margin-right: auto;"> <tbody> <tr> <td style="text-align: center;">2</td> </tr> </tbody> </table></td> <td>b^{3}a, <br>z<sub>4</sub>, z<sub>2</sub>, z<sub>5</sub>(z<sub>1</sub>, z<sub>3</sub>)</td> <td style="font-weight: bold;"><a style="font-weight: normal;" href="S8files/S8mod2b2.html">Condensation Information</a><br> <a style="font-weight: normal;" href="S8files/S8mod2b2extdim.html">Ext Dimensions</a><br> <a style="font-weight: normal;" href="S8files/S8mod2b2.gap.gz">Basic Algebra</a></td> </tr> <tr> <td colspan="1" rowspan="3">3</td> <td>1</td> <td>186</td><td><table style="text-align: left; margin-left: auto; margin-right: auto;"> <tbody> <tr> <td style="text-align: center;">4</td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;">4</td> <td style="text-align: center;">2</td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">4</td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;">3</td> <td style="text-align: center;">0</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">0</td> <td style="text-align: center;">3</td> </tr> </tbody> </table></td> <td>z<sub>7</sub>, z<sub>3</sub>, z<sub>4</sub>,<br>z<sub>7</sub>(z<sub>1</sub>, z<sub>4</sub>)</td> <td style="font-weight: bold;"><a style="font-weight: normal;" href="S8files/S8mod3b1.html">Condensation Information</a><br> <a style="font-weight: normal;" href="S8files/S8mod3b1extdim.html">Ext Dimensions</a><br> <a style="font-weight: normal;" href="S8files/S8mod3b1.gap.gz">Basic Algebra</a></td> </tr> <tr> <td>2</td> <td>186</td><td><table style="text-align: left; margin-left: auto; margin-right: auto;"> <tbody> <tr> <td style="text-align: center;">4</td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">4</td> <td style="text-align: center;">2</td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;">2</td> <td style="text-align: center;">4</td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;">3</td> <td style="text-align: center;">0</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">0</td> <td style="text-align: center;">3</td> </tr> </tbody> </table></td> <td>z<sub>7</sub>, z<sub>3</sub>, z<sub>4</sub>,<br>z<sub>7</sub>(z<sub>1</sub>, z<sub>4</sub>)</td> <td style="font-weight: bold;"><a style="font-weight: normal;" href="S8files/S8mod3b2.html">Condensation Information</a><br> <a style="font-weight: normal;" href="S8files/S8mod3b2extdim.html">Ext Dimensions</a><br> <a style="font-weight: normal;" href="S8files/S8mod3b2.gap.gz">Basic Algebra</a></td> </tr> <tr> <td>3</td> <td>68</td><td><table style="text-align: left; margin-left: auto; margin-right: auto;"> <tbody> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> </tr> </tbody> </table></td> <td>z<sub>8</sub>, z<sub>3</sub>, z<sub>2</sub>,<br>z<sub>6</sub>(z<sub>1</sub>, z<sub>3</sub>), z<sub>7</sub>(z<sub>1</sub>, z<sub>3</sub>)</td> <td style="font-weight: bold;"><a style="font-weight: normal;" href="S8files/S8mod3b3.html">Condensation Information</a><br> <a style="font-weight: normal;" href="S8files/S8mod3b3extdim.html">Ext Dimensions</a><br> <a style="font-weight: normal;" href="S8files/S8mod3b3.gap.gz">Basic Algebra</a></td> </tr> <tr> <td colspan="1" rowspan="3">5</td> <td>1</td> <td>60</td><td><table style="text-align: left; margin-left: auto; margin-right: auto;"> <tbody> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;">0</td> <td style="text-align: center;">0</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">0</td> <td style="text-align: center;">1</td> </tr> <tr> <td style="text-align: center;">0</td> <td style="text-align: center;">0</td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> </tr> <tr> <td style="text-align: center;">0</td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> </tr> </tbody> </table></td> <td>z<sub>2</sub>, z<sub>11</sub>, z<sub>8</sub></td> <td style="font-weight: bold;"><a style="font-weight: normal;" href="S8files/S8mod5b1.html">Condensation Information</a><br> <a style="font-weight: normal;" href="S8files/S8mod5b1extdim.html">Ext Dimensions</a><br> <a style="font-weight: normal;" href="S8files/S8mod5b1.gap.gz">Basic Algebra</a></td> </tr> <tr> <td>2</td> <td>60</td><td><table style="text-align: left; margin-left: auto; margin-right: auto;"> <tbody> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;">0</td> <td style="text-align: center;">0</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">0</td> <td style="text-align: center;">1</td> </tr> <tr> <td style="text-align: center;">0</td> <td style="text-align: center;">0</td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> </tr> <tr> <td style="text-align: center;">0</td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> </tr> </tbody> </table></td> <td>z<sub>2</sub>, z<sub>11</sub>, z<sub>8</sub></td> <td style="font-weight: bold;"><a style="font-weight: normal;" href="S8files/S8mod5b2.html">Condensation Information</a><br> <a style="font-weight: normal;" href="S8files/S8mod5b2extdim.html">Ext Dimensions</a><br> <a style="font-weight: normal;" href="S8files/S8mod5b2.gap.gz">Basic Algebra</a></td> </tr> <tr> <td>3</td> <td>72</td><td><table style="text-align: left; margin-left: auto; margin-right: auto;"> <tbody> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">0</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">0</td> <td style="text-align: center;">1</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">0</td> <td style="text-align: center;">2</td> <td style="text-align: center;">0</td> </tr> <tr> <td style="text-align: center;">0</td> <td style="text-align: center;">1</td> <td style="text-align: center;">0</td> <td style="text-align: center;">2</td> </tr> </tbody> </table></td> <td>z<sub>11</sub>, z<sub>8</sub></td> <td style="font-weight: bold;"><a style="font-weight: normal;" href="S8files/S8mod5b3.html">Condensation Information</a><br> <a style="font-weight: normal;" href="S8files/S8mod5b3extdim.html">Ext Dimensions</a><br> <a style="font-weight: normal;" href="S8files/S8mod5b3.gap.gz">Basic Algebra</a></td> </tr> <tr> <td colspan="1" rowspan="1">7</td> <td>1</td> <td>34</td><td><table style="text-align: left; margin-left: auto; margin-right: auto;"> <tbody> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;">0</td> <td style="text-align: center;">0</td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">0</td> </tr> <tr> <td style="text-align: center;">0</td> <td style="text-align: center;">2</td> <td style="text-align: center;">0</td> <td style="text-align: center;">0</td> <td style="text-align: center;">1</td> <td style="text-align: center;">0</td> </tr> <tr> <td style="text-align: center;">0</td> <td style="text-align: center;">0</td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;">0</td> <td style="text-align: center;">1</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">0</td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">0</td> <td style="text-align: center;">0</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">0</td> <td style="text-align: center;">0</td> <td style="text-align: center;">2</td> <td style="text-align: center;">0</td> </tr> <tr> <td style="text-align: center;">0</td> <td style="text-align: center;">0</td> <td style="text-align: center;">1</td> <td style="text-align: center;">0</td> <td style="text-align: center;">0</td> <td style="text-align: center;">2</td> </tr> </tbody> </table></td> <td>z<sub>3</sub>, z<sub>7</sub>, z<sub>11</sub>,<br>z<sub>4</sub></td> <td style="font-weight: bold;"><a style="font-weight: normal;" href="S8files/S8mod7b1.html">Condensation Information</a><br> <a style="font-weight: normal;" href="S8files/S8mod7b1extdim.html">Ext Dimensions</a><br> <a style="font-weight: normal;" href="S8files/S8mod7b1.gap.gz">Basic Algebra</a></td> </tr> </tbody> </table> </body> </html>