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Each Galois group is identified by its degree and an index called its T-number. This specifies both the abstract group and a faithful transitive permutation of that group. One may search for a group in the form $n$T$t$ where $n$ is the degree and $t$ is the T-number. The search results will then only show fields where the Galois group matches the requested permutation representation.

For familiar groups one can use short names from the table below. An abstract group may have more than one representation as a Galois group. Correspondingly, the familiar symbol for a group may represent several Galois groups. When searching for fields using the name of an abstract group, e.g., S3, the results may contain fields with different degrees and permutation representations. Some of these combinations are shown below for $n\leq 47$.

<table border=1 cellpadding=5 class="right_align_table"><thead><tr><th>Alias</th><th>Group</th><th>\(n\)T\(t\)</th></tr></thead><tbody><tr><td>V4</td><td>$C_2^2$</td><td>4T2</td></tr><tr><td>C2XC2</td><td>$C_2^2$</td><td>4T2</td></tr><tr><td>F5</td><td>$F_5$</td><td>5T3, 10T4, 20T5</td></tr><tr><td>PSL(2,5)</td><td>$A_5$</td><td>5T4, 6T12, 10T7, 12T33, 15T5, 20T15, 30T9</td></tr><tr><td>PGL(2,5)</td><td>$S_5$</td><td>5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62</td></tr><tr><td>S3XC3</td><td>$S_3\times C_3$</td><td>6T5, 9T4, 18T3</td></tr><tr><td>S3XS3</td><td>$S_3^2$</td><td>6T9, 9T8, 12T16, 18T9, 18T11, 36T13</td></tr><tr><td>PSL(2,9)</td><td>$A_6$</td><td>6T15, 10T26, 15T20, 20T89, 30T88, 36T555, 40T304, 45T49</td></tr><tr><td>F7</td><td>$F_7$</td><td>7T4, 14T4, 21T4, 42T4</td></tr><tr><td>GL(3,2)</td><td>$\GL(3,2)$</td><td>7T5, 8T37, 14T10, 21T14, 24T284, 28T32, 42T37, 42T38</td></tr><tr><td>PSL(2,7)</td><td>$\GL(3,2)$</td><td>7T5, 8T37, 14T10, 21T14, 24T284, 28T32, 42T37, 42T38</td></tr><tr><td>C4XC2</td><td>$C_4\times C_2$</td><td>8T2</td></tr><tr><td>C2XC2XC2</td><td>$C_2^3$</td><td>8T3</td></tr><tr><td>Q8</td><td>$Q_8$</td><td>8T5</td></tr><tr><td>SL(2,3)</td><td>$\SL(2,3)$</td><td>8T12, 24T7</td></tr><tr><td>GL(2,3)</td><td>$\textrm{GL(2,3)}$</td><td>8T23, 16T66, 24T22</td></tr><tr><td>PGL(2,7)</td><td>$\PGL(2,7)$</td><td>8T43, 14T16, 16T713, 21T20, 24T707, 28T42, 28T46, 42T81, 42T82, 42T83</td></tr><tr><td>C3XC3</td><td>$C_3^2$</td><td>9T2</td></tr><tr><td>M9</td><td>$C_3^2:Q_8$</td><td>9T14, 12T47, 18T35, 24T82, 36T55</td></tr><tr><td>PSL(2,8)</td><td>$\PSL(2,8)$</td><td>9T27, 28T70, 36T712</td></tr><tr><td>PGL(2,9)</td><td>$\PGL(2,9)$</td><td>10T30, 12T182, 20T146, 30T171, 36T1254, 40T590, 45T110</td></tr><tr><td>M10</td><td>$M_{10}$</td><td>10T31, 12T181, 20T148, 20T150, 30T162, 36T1253, 40T591, 45T109</td></tr><tr><td>F11</td><td>$F_{11}$</td><td>11T4, 22T4</td></tr><tr><td>PSL(2,11)</td><td>$\PSL(2,11)$</td><td>11T5, 12T179</td></tr><tr><td>M11</td><td>$M_{11}$</td><td>11T6, 12T272, 22T22</td></tr><tr><td>C6XC2</td><td>$C_6\times C_2$</td><td>12T2</td></tr><tr><td>C3:C4</td><td>$C_3 : C_4$</td><td>12T5</td></tr><tr><td>M12</td><td>$M_{12}$</td><td>12T295</td></tr><tr><td>F13</td><td>$F_{13}$</td><td>13T6, 26T8, 39T11</td></tr><tr><td>PSL(3,3)</td><td>$\PSL(3,3)$</td><td>13T7, 26T39, 39T43</td></tr><tr><td>PSL(2,13)</td><td>$\PSL(2,13)$</td><td>14T30, 28T120, 42T176</td></tr><tr><td>PGL(2,13)</td><td>$\PGL(2,13)$</td><td>14T39, 28T201, 42T284</td></tr><tr><td>Q8XC2</td><td>$Q_8\times C_2$</td><td>16T7</td></tr><tr><td>C4:C4</td><td>$C_4:C_4$</td><td>16T8</td></tr><tr><td>Q16</td><td>$Q_{16}$</td><td>16T14</td></tr><tr><td>F17</td><td>$F_{17}$</td><td>17T5</td></tr><tr><td>PSL(2,17)</td><td>$\PSL(2,16)$</td><td>17T6</td></tr><tr><td>PGL(2,17)</td><td>$\PGL(2,17)$</td><td>18T468</td></tr><tr><td>C5:C4</td><td>$C_5:C_4$</td><td>20T2</td></tr><tr><td>PGL(2,19)</td><td>t20n362</td><td>20T362, 40T5409</td></tr><tr><td>M22</td><td>$M_{22}$</td><td>22T38</td></tr><tr><td>F23</td><td>$C_{23}:C_{11}$</td><td>23T3</td></tr><tr><td>M23</td><td>$M_{23}$</td><td>23T5</td></tr><tr><td>Q8XC3</td><td>$C_3\times Q_8$</td><td>24T4</td></tr><tr><td>C3:Q8</td><td>$C_3:Q_8$</td><td>24T5</td></tr><tr><td>C3:C8</td><td>$C_3:C_8$</td><td>24T8</td></tr><tr><td>M24</td><td>$M_{24}$</td><td>24T24680</td></tr><tr><td>PSP(4,3)</td><td>$\PSp(4,3)$</td><td>27T993, 36T12781, 40T14344, 40T14345, 45T666</td></tr><tr><td>C7:C4</td><td>$C_7:C_4$</td><td>28T3</td></tr><tr><td>PSU(3,3)</td><td>$\PSU(3,3)$</td><td>28T323, 36T6815</td></tr><tr><td>Q32</td><td>$Q_{32}$</td><td>32T51</td></tr><tr><td>C5:C8</td><td>$C_5:C_8$</td><td>40T3</td></tr></tbody></table>

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  • Review status: reviewed
  • Last edited by John Jones on 2019-09-06 18:36:54
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