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Aliases
Splittings
Subgroups
18 conjugacy classes of subgroups.
Full subgroup data →
Presentation
A minimal faithful permutation representation — 2 generators on 2146 points:
(2,3)(5,6,8)(7,10,15)(9,13,12)(11,17,25)(14,20,22)(16,23,19)(18,27,38)(21,30,35)(24,33,32)(26,36,29)(28,40,54)(31,43,51)(34,46,48)(37,49,45)(39,52,42)(41,56,73)(44,59,70)(47,62,67)(50,65,64)(53,68,61)(55,71,58)(57,75,95)(60,78,92)(63,81,89)(66,84,86)(69,87,83)(72,90,80)(74,93,77)(76,97,120)(79,100,117)(82,103,114)(85,106,111)(88,109,108)(91,112,105)(94,115,102)(96,118,99)(98,122,148)(101,125,145)(104,128,142)(107,131,139)(110,134,136)(113,137,133)(116,140,130)(119,143,127)(121,146,124)(123,150,179)(126,153,176)(129,156,173)(132,159,170)(135,162,167)(138,165,164)(141,168,161)(144,171,158)(147,174,155)(149,177,152)(151,181,213)(154,184,210)(157,187,207)(160,190,204)(163,193,201)(166,196,198)(169,199,195)(172,202,192)(175,205,189)(178,208,186)(180,211,183)(182,215,250)(185,218,247)(188,221,244)(191,224,241)(194,227,238)(197,230,235)(200,233,232)(203,236,229)(206,239,226)(209,242,223)(212,245,220)(214,248,217)(216,252,290)(219,255,287)(222,258,284)(225,261,281)(228,264,278)(231,267,275)(234,270,272)(237,273,269)(240,276,266)(243,279,263)(246,282,260)(249,285,257)(251,288,254)(253,292,333)(256,295,330)(259,298,327)(262,301,324)(265,304,321)(268,307,318)(271,310,315)(274,313,312)(277,316,309)(280,319,306)(283,322,303)(286,325,300)(289,328,297)(291,331,294)(293,335,379)(296,338,376)(299,341,373)(302,344,370)(305,347,367)(308,350,364)(311,353,361)(314,356,358)(317,359,355)(320,362,352)(323,365,349)(326,368,346)(329,371,343)(332,374,340)(334,377,337)(336,381,428)(339,384,425)(342,387,422)(345,390,419)(348,393,416)(351,396,413)(354,399,410)(357,402,407)(360,405,404)(363,408,401)(366,411,398)(369,414,395)(372,417,392)(375,420,389)(378,423,386)(380,426,383)(382,430,480)(385,433,477)(388,436,474)(391,439,471)(394,442,468)(397,445,465)(400,448,462)(403,451,459)(406,454,456)(409,457,453)(412,460,450)(415,463,447)(418,466,444)(421,469,441)(424,472,438)(427,475,435)(429,478,432)(431,482,535)(434,485,532)(437,488,529)(440,491,526)(443,494,523)(446,497,520)(449,500,517)(452,503,514)(455,506,511)(458,509,508)(461,512,505)(464,515,502)(467,518,499)(470,521,496)(473,524,493)(476,527,490)(479,530,487)(481,533,484)(483,537,593)(486,540,590)(489,543,587)(492,546,584)(495,549,581)(498,552,578)(501,555,575)(504,558,572)(507,561,569)(510,564,566)(513,567,563)(516,570,560)(519,573,557)(522,576,554)(525,579,551)(528,582,548)(531,585,545)(534,588,542)(536,591,539)(538,595,654)(541,598,651)(544,601,648)(547,604,645)(550,607,642)(553,610,639)(556,613,636)(559,616,633)(562,619,630)(565,622,627)(568,625,624)(571,628,621)(574,631,618)(577,634,615)(580,637,612)(583,640,609)(586,643,606)(589,646,603)(592,649,600)(594,652,597)(596,656,718)(599,659,715)(602,662,712)(605,665,709)(608,668,706)(611,671,703)(614,674,700)(617,677,697)(620,680,694)(623,683,691)(626,686,688)(629,689,685)(632,692,682)(635,695,679)(638,698,676)(641,701,673)(644,704,670)(647,707,667)(650,710,664)(653,713,661)(655,716,658)(657,720,785)(660,723,782)(663,726,779)(666,729,776)(669,732,773)(672,735,770)(675,738,767)(678,741,764)(681,744,761)(684,747,758)(687,750,755)(690,753,752)(693,756,749)(696,759,746)(699,762,743)(702,765,740)(705,768,737)(708,771,734)(711,774,731)(714,777,728)(717,780,725)(719,783,722)(721,787,855)(724,790,852)(727,793,849)(730,796,846)(733,799,843)(736,802,840)(739,805,837)(742,808,834)(745,811,831)(748,814,828)(751,817,825)(754,820,822)(757,823,819)(760,826,816)(763,829,813)(766,832,810)(769,835,807)(772,838,804)(775,841,801)(778,844,798)(781,847,795)(784,850,792)(786,853,789)(788,857,928)(791,860,925)(794,863,922)(797,866,919)(800,869,916)(803,872,913)(806,875,910)(809,878,907)(812,881,904)(815,884,901)(818,887,898)(821,890,895)(824,893,892)(827,896,889)(830,899,886)(833,902,883)(836,905,880)(839,908,877)(842,911,874)(845,914,871)(848,917,868)(851,920,865)(854,923,862)(856,926,859)(858,930,1004)(861,933,1001)(864,936,998)(867,939,995)(870,942,992)(873,945,989)(876,948,986)(879,951,983)(882,954,980)(885,957,977)(888,960,974)(891,963,971)(894,966,968)(897,969,965)(900,972,962)(903,975,959)(906,978,956)(909,981,953)(912,984,950)(915,987,947)(918,990,944)(921,993,941)(924,996,938)(927,999,935)(929,1002,932)(931,1006,1083)(934,1009,1080)(937,1012,1077)(940,1015,1074)(943,1018,1071)(946,1021,1068)(949,1024,1065)(952,1027,1062)(955,1030,1059)(958,1033,1056)(961,1036,1053)(964,1039,1050)(967,1042,1047)(970,1045,1044)(973,1048,1041)(976,1051,1038)(979,1054,1035)(982,1057,1032)(985,1060,1029)(988,1063,1026)(991,1066,1023)(994,1069,1020)(997,1072,1017)(1000,1075,1014)(1003,1078,1011)(1005,1081,1008)(1007,1085,1165)(1010,1088,1162)(1013,1091,1159)(1016,1094,1156)(1019,1097,1153)(1022,1100,1150)(1025,1103,1147)(1028,1106,1144)(1031,1109,1141)(1034,1112,1138)(1037,1115,1135)(1040,1118,1132)(1043,1121,1129)(1046,1124,1126)(1049,1127,1123)(1052,1130,1120)(1055,1133,1117)(1058,1136,1114)(1061,1139,1111)(1064,1142,1108)(1067,1145,1105)(1070,1148,1102)(1073,1151,1099)(1076,1154,1096)(1079,1157,1093)(1082,1160,1090)(1084,1163,1087)(1086,1167,1250)(1089,1170,1247)(1092,1173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130,972,826,692,570,460,362,276,202,140,90,52,25,35,48,64,83,105,130,158,189,223,260,300,343,389,438,490,545,603,664,728,795,865,938,1014,1093,1175,1260,1348,1436,1521,1600,1632,1468,1291,1121,963,817,683,561,451,353,267,193,131,81,43,17,26,37,50,66,85,107,132,160,191,225,262,302,345,391,440,492,547,605,666,730,797,867,940,1016,1095,1177,1262,1350,1438,1523,1602,1461,1282,1112,954,808,674,552,442,344,258,184,122,149,178,209,243,280,320,363,409,458,510,565,623,684,748,815,885,958,1034,1113,1195,1280,1368,1456,1541,1620,1693,1760,1821,1876,1925,1968,1881,1762,1619,1452,1273,1103,945,799,665,543,433,335,380,427,476,528,583,641,702,766,833,903,976,1052,1131,1213,1298,1385,1472,1555,1598,1426,1250,1335,1422,1509,1591,1667,1737,1801,1859,1911,1957,1997,2031,1967,1872,1753,1610,1443,1264,1094,936,790,656,719,784,851,921,994,1070,1149,1231,1316,1403,1490,1572,1648,1718,1783,1842,1895,1942,1964,1863,1738,1589,1417,1242,1078,926,785,852,922,995,1071,1150,1232,1317,1404,1491,1573,1649,1719,1601,1434,1255,1085,1166,1249,1334,1421,1508,1590,1666,1736,1800,1858,1910,1956,1996,2030,2059,2082,2100,2114,2119,2085,2032,1955,1854,1729,1580,1408,1233,1069,917,777,649,533,428,477,529,584,642,703,767,834,904,977,1053,1132,1214,1299,1386,1473,1556,1633,1705,1772,1833,1888,1937,1980,2017,2048,2073,2093,2109,2122,2132,2135,2113,2079,2025,1947,1845,1720,1571,1399,1224,1060,908,768,640,524,420,328,248,179,210,244,281,321,364,410,459,511,566,624,685,749,816,886,959,1035,1114,1196,1281,1369,1457,1542,1621,1694,1761,1822,1877,1926,1969,2006,2020,1941,1839,1713,1563,1390,1215,1051,899,759,631,515,411,319,239,171,115,71,38,51,67,86,108,133,161,192,226,263,303,346,392,441,493,548,606,667,731,798,868,941,1017,1096,1178,1263,1351,1439,1524,1603,1676,1706,1554,1381,1206,1042,890,750,622,506,402,310,230,162,106,62,30,10,16,24,34,47,63,82,104,129,157,188,222,259,299,342,388,437,489,544,602,663,727,794,864,937,1013,1092,1174,1259,1347,1435,1520,1373,1197,1033,881,741,613,497,393,301,221,153,97,121,147,175,206,240,277,317,360,406,455,507,562,620,681,745,812,882,955,1031,1110,1192,1277,1365,1453,1538,1617,1690,1757,1818,1873,1922,1826,1695,1540,1364,1188,1024,872,732,604,488,384,292,334,378,424,473,525,580,638,699,763,830,900,973,1049,1128,1210,1295,1382,1469,1516,1338,1425,1512,1594,1670,1740,1804,1862,1914,1960,2000,2034,2062,2007,1924,1817,1686,1531,1355,1179,1015,863,723,595,655,717,781,848,918,991,1067,1146,1228,1313,1400,1487,1569,1645,1716,1781,1840,1893,1918,1805,1668,1507,1330,1160,1002,855,925,998,1074,1153,1235,1320,1407,1494,1576,1652,1722,1786,1677,1522,1346,1170,1006,1084,1164,1246,1331,1418,1505,1587,1663,1733,1797,1855,1907,1953,1993,2028,2057,2080,2098,2105,2063,1998,1909,1796,1659,1498,1321,1151,993,847,713,591,480,532,587,645,706,770,837,907,980,1056,1135,1217,1302,1389,1476,1559,1636,1707,1773,1834,1889,1938,1981,2018,2049,2074,2094,2110,2123,2133,2140,2142,2126,2099,2056,1990,1900,1787,1650,1489,1312,1142,984,838,704,582,472,374,288,213,247,284,324,367,413,462,514,569,627,688,752,819,889,962,1038,1117,1199,1284,1372,1460,1545,1624,1697,1764,1825,1880,1929,1972,2009,2040,2051,1984,1894,1780,1642,1480,1303,1133,975,829,695,573,463,365,279,205,143,93,54,70,89,111,136,164,195,229,266,306,349,395,444,496,551,609,670,734,801,871,944,1020,1099,1181,1266,1354,1442,1527,1606,1679,1746,1774,1634,1471,1294,1124,966,820,686,564,454,356,270,196,134,84,46,20,6,9,14,21,31,44,60,79,101,126,154,185,219,256,296,339,385,434,486,541,599,660,724,791,861,934,1010,1089,1171,1256,1344,1432,1285,1115,957,811,677,555,445,347,261,187,125,75,96,119,144,172,203,237,274,314,357,403,452,504,559,617,678,742,809,879,952,1028,1107,1189,1274,1362,1450,1535,1614,1687,1754,1815,1870,1765,1622,1455,1276,1106,948,802,668,546,436,338,252,291,332,375,421,470,522,577,635,696,760,827,897,970,1046,1125,1207,1292,1379,1428,1515,1597,1673,1743,1807,1865,1917,1963,2003,2037,2065,2087,2041,1970,1875,1756,1613,1446,1267,1097,939,793,659,537,594,653,714,778,845,915,988,1064,1143,1225,1310,1397,1484,1566,1643,1714,1779,1838,1866,1741,1592,1420,1245,1081,928,1001,1077,1156,1238,1323,1410,1497,1579,1655,1725,1789,1847,1747,1604,1437,1258,1088,930,1005,1082,1161,1243,1328,1415,1502,1584,1660,1730,1794,1852,1904,1950,1991,2026,2055,2078,2088,2035,1958,1857,1732,1583,1411,1236,1072,920,780,652,535,590,648,709,773,840,910,983,1059,1138,1220,1305,1392,1479,1562,1639,1710,1775,1835,1890,1939,1982,2019,2050,2075,2095,2111,2124,2134,2141,2145,2146,2136,2115,2081,2027,1949,1848,1723,1574,1402,1227,1063,911,771,643,527,423,331,250,287,327,370,416,465,517,572,630,691,755,822,892,965,1041,1120,1202,1287,1375,1463,1548,1627,1700,1767,1828,1883,1932,1975,2012,2043,2068,2076,2021,1943,1841,1715,1565,1393,1218,1054,902,762,634,518,414,322,242,174,118,73,92,114,139,167,198,232,269,309,352,398,447,499,554,612,673,737,804,874,947,1023,1102,1184,1269,1357,1445,1530,1609,1682,1749,1810,1836,1708,1557,1384,1209,1045,893,753,625,509,405,313,233,165,109,65,33,13)
Conjugacy classes & character table
2151 conjugacy classes. The character table is too large to display.
GAP
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38574, 12);
Finite Group Databasebeta