Finite Group DatabasebetaFinite Group Databasebeta| order | 28206 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(D1567⋊C3) |
| unambiguous semidirect product | (C1567⋊C3)⋊D3 |
| unambiguous semidirect product | D4701⋊C3 |
| Fitting, socle, derived | C3×C1567 ⋊ C2×C3 |
| Hall | C3×C1567⋊C3 ⋊ C2 |
| Sylow | C1567 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C1567 | C2×C3 |
| LCS | G ▷ C3×C1567 | |
| UCS | C1 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1570 |
| transitive degree | μt(G) = 4701 |
A minimal faithful permutation representation — 2 generators on 1570 points:
(2,3)(5,9,10,6,7,11)(8,17,18,12,13,27)(14,30,31,20,21,44)(15,29,25,23,19,16)(22,49,50,33,34,67)(24,47,48,36,32,41)(26,28,46,38,39,42)(35,72,73,52,53,96)(37,70,71,55,51,64)(40,45,69,57,58,65)(43,85,83,61,62,60)(54,101,102,75,76,131)(56,99,100,78,74,93)(59,68,98,80,81,94)(63,114,115,86,87,89)(66,120,112,90,91,84)(77,136,137,104,105,172)(79,134,135,107,103,128)(82,97,133,109,110,129)(88,149,150,116,117,124)(92,156,154,121,122,119)(95,161,147,125,126,113)(106,177,178,139,140,219)(108,175,176,142,138,169)(111,132,174,144,145,170)(118,190,191,151,152,165)(123,197,198,157,158,160)(127,203,195,162,163,155)(130,208,188,166,167,148)(141,224,225,180,181,272)(143,222,223,183,179,216)(146,173,221,185,186,217)(153,237,238,192,193,212)(159,244,245,199,200,207)(164,251,249,204,205,202)(168,256,242,209,210,196)(171,261,235,213,214,189)(182,277,278,227,228,331)(184,275,276,230,226,269)(187,220,274,232,233,270)(194,290,291,239,240,265)(201,297,298,246,247,260)(206,304,305,252,253,255)(211,310,302,257,258,250)(215,315,295,262,263,243)(218,320,288,266,267,236)(229,336,337,280,281,396)(231,334,335,283,279,328)(234,273,333,285,286,329)(241,349,350,292,293,324)(248,356,357,299,300,319)(254,363,364,306,307,314)(259,370,368,311,312,309)(264,375,361,316,317,303)(268,380,354,321,322,296)(271,385,347,325,326,289)(282,401,402,339,340,467)(284,399,400,342,338,393)(287,332,398,344,345,394)(294,414,415,351,352,389)(301,421,422,358,359,384)(308,428,429,365,366,379)(313,435,436,371,372,374)(318,441,433,376,377,369)(323,446,426,381,382,362)(327,451,419,386,387,355)(330,456,412,390,391,348)(341,472,473,404,405,544)(343,470,471,407,403,464)(346,397,469,409,410,465)(353,485,486,416,417,460)(360,492,493,423,424,455)(367,499,500,430,431,450)(373,506,507,437,438,445)(378,513,511,442,443,440)(383,518,504,447,448,434)(388,523,497,452,453,427)(392,528,490,457,458,420)(395,533,483,461,462,413)(406,549,550,475,476,627)(408,547,548,478,474,541)(411,468,546,480,481,542)(418,562,563,487,488,537)(425,569,570,494,495,532)(432,576,577,501,502,527)(439,583,584,508,509,522)(444,590,591,514,515,517)(449,596,588,519,520,512)(454,601,581,524,525,505)(459,606,574,529,530,498)(463,611,567,534,535,491)(466,616,560,538,539,484)(477,632,633,552,553,716)(479,630,631,555,551,624)(482,545,629,557,558,625)(489,645,646,564,565,620)(496,652,653,571,572,615)(503,659,660,578,579,610)(510,666,667,585,586,605)(516,673,674,592,593,600)(521,680,678,597,598,595)(526,685,671,602,603,589)(531,690,664,607,608,582)(536,695,657,612,613,575)(540,700,650,617,618,568)(543,705,643,621,622,561)(554,721,722,635,636,811)(556,719,720,638,634,713)(559,628,718,640,641,714)(566,734,735,647,648,709)(573,741,742,654,655,704)(580,748,749,661,662,699)(587,755,756,668,669,694)(594,762,763,675,676,689)(599,769,770,681,682,684)(604,775,767,686,687,679)(609,780,760,691,692,672)(614,785,753,696,697,665)(619,790,746,701,702,658)(623,795,739,706,707,651)(626,800,732,710,711,644)(637,816,817,724,725,912)(639,814,815,727,723,808)(642,717,813,729,730,809)(649,829,830,736,737,804)(656,836,837,743,744,799)(663,843,844,750,751,794)(670,850,851,757,758,789)(677,857,858,764,765,784)(683,864,865,771,772,779)(688,871,869,776,777,774)(693,876,862,781,782,768)(698,881,855,786,787,761)(703,886,848,791,792,754)(708,891,841,796,797,747)(712,896,834,801,802,740)(715,901,827,805,806,733)(726,917,918,819,820,1019)(728,915,916,822,818,909)(731,812,914,824,825,910)(738,930,931,831,832,905)(745,937,938,838,839,900)(752,944,945,845,846,895)(759,951,952,852,853,890)(766,958,959,859,860,885)(773,965,966,866,867,880)(778,972,973,872,873,875)(783,978,970,877,878,870)(788,983,963,882,883,863)(793,988,956,887,888,856)(798,993,949,892,893,849)(803,998,942,897,898,842)(807,1003,935,902,903,835)(810,1008,928,906,907,828)(821,1024,1025,920,921,1132)(823,1022,1023,923,919,1016)(826,913,1021,925,926,1017)(833,1037,1038,932,933,1012)(840,1044,1045,939,940,1007)(847,1051,1052,946,947,1002)(854,1058,1059,953,954,997)(861,1065,1066,960,961,992)(868,1072,1073,967,968,987)(874,1079,1080,974,975,982)(879,1086,1084,979,980,977)(884,1091,1077,984,985,971)(889,1096,1070,989,990,964)(894,1101,1063,994,995,957)(899,1106,1056,999,1000,950)(904,1111,1049,1004,1005,943)(908,1116,1042,1009,1010,936)(911,1121,1035,1013,1014,929)(922,1137,1138,1027,1028,1246)(924,1135,1136,1030,1026,1129)(927,1020,1134,1032,1033,1130)(934,1149,1150,1039,1040,1125)(941,1154,1155,1046,1047,1120)(948,1161,1162,1053,1054,1115)(955,1168,1169,1060,1061,1110)(962,1175,1176,1067,1068,1105)(969,1182,1183,1074,1075,1100)(976,1189,1190,1081,1082,1095)(981,1196,1197,1087,1088,1090)(986,1202,1194,1092,1093,1085)(991,1207,1187,1097,1098,1078)(996,1212,1180,1102,1103,1071)(1001,1217,1173,1107,1108,1064)(1006,1222,1166,1112,1113,1057)(1011,1227,1159,1117,1118,1050)(1015,1232,1141,1122,1123,1043)(1018,1237,1147,1126,1127,1036)(1029,1238,1239,1140,1128,1339)(1031,1248,1249,1142,1139,1244)(1034,1133,1247,1144,1145,1245)(1041,1252,1253,1151,1152,1241)(1048,1257,1258,1156,1157,1236)(1055,1262,1263,1163,1164,1231)(1062,1269,1270,1170,1171,1226)(1069,1276,1277,1177,1178,1221)(1076,1283,1284,1184,1185,1216)(1083,1290,1291,1191,1192,1211)(1089,1297,1298,1198,1199,1206)(1094,1304,1302,1203,1204,1201)(1099,1309,1295,1208,1209,1195)(1104,1314,1288,1213,1214,1188)(1109,1319,1281,1218,1219,1181)(1114,1324,1274,1223,1224,1174)(1119,1329,1267,1228,1229,1167)(1124,1334,1251,1233,1234,1160)(1131,1343,1146,1242,1243,1148)(1143,1340,1341,1250,1240,1345)(1153,1346,1347,1254,1255,1344)(1158,1348,1349,1259,1260,1342)(1165,1352,1353,1264,1265,1338)(1172,1357,1358,1271,1272,1333)(1179,1362,1363,1278,1279,1328)(1186,1369,1370,1285,1286,1323)(1193,1376,1377,1292,1293,1318)(1200,1383,1384,1299,1300,1313)(1205,1390,1391,1305,1306,1308)(1210,1396,1388,1310,1311,1303)(1215,1401,1381,1315,1316,1296)(1220,1406,1374,1320,1321,1289)(1225,1411,1367,1325,1326,1282)(1230,1416,1351,1330,1331,1275)(1235,1421,1256,1335,1336,1268)(1261,1422,1423,1350,1337,1426)(1266,1427,1428,1354,1355,1425)(1273,1429,1430,1359,1360,1424)(1280,1433,1434,1364,1365,1420)(1287,1438,1439,1371,1372,1415)(1294,1443,1444,1378,1379,1410)(1301,1450,1451,1385,1386,1405)(1307,1457,1458,1392,1393,1400)(1312,1464,1462,1397,1398,1395)(1317,1469,1455,1402,1403,1389)(1322,1474,1448,1407,1408,1382)(1327,1479,1432,1412,1413,1375)(1332,1484,1356,1417,1418,1368)(1361,1485,1486,1431,1419,1489)(1366,1490,1491,1435,1436,1488)(1373,1492,1493,1440,1441,1487)(1380,1496,1497,1445,1446,1483)(1387,1501,1502,1452,1453,1478)(1394,1506,1507,1459,1460,1473)(1399,1513,1514,1465,1466,1468)(1404,1519,1511,1470,1471,1463)(1409,1524,1495,1475,1476,1456)(1414,1529,1437,1480,1481,1449)(1442,1530,1531,1494,1482,1534)(1447,1535,1536,1498,1499,1533)(1454,1537,1538,1503,1504,1532)(1461,1541,1542,1508,1509,1528)(1467,1546,1547,1515,1516,1523)(1472,1551,1540,1520,1521,1518)(1477,1556,1500,1525,1526,1512)(1505,1557,1558,1539,1527,1561)(1510,1562,1563,1543,1544,1560)(1517,1564,1565,1548,1549,1559)(1522,1567,1545,1552,1553,1555)(1550,1568,1569,1566,1554,1570);(1,2,3)(4,5,8,14,22,35,54,77,106,141,182,229,282,341,406,477,554,637,726,821,922,1029,1141,1042,935,834,739,650,567,490,419,354,295,242,195,154,115,71,31,46,47,30,45,66,92,123,159,201,248,301,360,425,496,573,656,745,840,941,1048,1158,1261,1351,1274,1173,1063,956,855,760,671,588,511,436,364,298,238,176,102,133,161,156,149,134,101,132,171,215,264,318,378,444,516,594,677,766,861,962,1069,1179,1280,1366,1437,1448,1381,1295,1194,1084,973,865,763,667,577,493,415,335,225,274,320,315,310,304,297,290,275,224,273,330,392,459,531,609,693,783,879,981,1089,1200,1301,1387,1454,1505,1540,1514,1458,1384,1291,1183,1066,952,844,742,646,548,402,469,533,528,523,518,513,506,499,492,485,470,401,468,543,623,708,798,894,996,1104,1215,1317,1404,1472,1522,1554,1569,1563,1538,1497,1439,1363,1270,1162,1045,931,815,633,718,800,795,790,785,780,775,769,762,755,748,741,734,719,632,717,810,908,1011,1119,1230,1332,1419,1436,1441,1446,1453,1460,1466,1471,1476,1481,1486,1428,1349,1253,1136,918,1021,1121,1116,1111,1106,1101,1096,1091,1086,1079,1072,1065,1058,1051,1044,1037,1022,917,1020,1131,1028,1139,1152,1157,1164,1171,1178,1185,1192,1199,1204,1209,1214,1219,1224,1229,1234,1239,1243,1245,1246,1345,1421,1416,1411,1406,1401,1396,1390,1383,1376,1369,1362,1357,1352,1348,1346,1340,1238,1126,926,725,818,832,839,846,853,860,867,873,878,883,888,893,898,903,907,910,912,1016,1125,1236,1338,1424,1488,1534,1556,1551,1546,1541,1537,1535,1530,1480,1412,1325,1223,1112,1004,902,805,641,476,551,565,572,579,586,593,598,603,608,613,618,622,625,627,713,804,900,1002,1110,1221,1323,1410,1478,1528,1559,1566,1548,1515,1465,1397,1310,1208,1097,989,887,791,701,617,538,410,281,338,352,359,366,372,377,382,387,391,394,396,464,537,615,699,789,885,987,1095,1206,1308,1395,1463,1512,1498,1445,1378,1292,1191,1081,974,872,776,686,602,524,452,386,325,233,140,179,193,200,205,210,214,217,219,269,324,384,450,522,600,684,774,870,971,1078,1188,1289,1375,1431,1359,1271,1170,1060,953,852,757,668,585,508,437,371,311,257,209,166,110,53,74,87,91,94,96,128,165,207,255,309,369,434,505,582,665,754,849,950,1057,1167,1268,1254,1151,1039,932,831,736,647,564,487,416,351,292,239,192,151,116,86,61,39,13,19,27,41,60,84,113,148,189,236,289,348,413,484,561,644,733,828,929,1036,1148,1144,1032,925,824,729,640,557,480,409,344,285,232,185,144,109,80,57,38,25,10,9,15,24,37,56,79,108,143,184,231,284,343,408,479,556,639,728,823,924,1031,1143,1251,1159,1049,942,841,746,657,574,497,426,361,302,249,198,150,100,50,69,85,70,49,68,95,127,164,206,254,308,367,432,503,580,663,752,847,948,1055,1165,1266,1356,1367,1281,1180,1070,963,862,767,678,591,507,429,357,291,223,137,174,208,203,197,190,175,136,173,218,268,323,383,449,521,599,683,773,868,969,1076,1186,1287,1373,1442,1495,1455,1388,1302,1197,1080,966,858,756,660,570,486,400,278,333,385,380,375,370,363,356,349,334,277,332,395,463,536,614,698,788,884,986,1094,1205,1307,1394,1461,1510,1545,1547,1507,1451,1377,1284,1176,1059,945,837,735,631,473,546,616,611,606,601,596,590,583,576,569,562,547,472,545,626,712,803,899,1001,1109,1220,1322,1409,1477,1527,1544,1549,1553,1558,1536,1493,1434,1358,1263,1155,1038,916,722,813,901,896,891,886,881,876,871,864,857,850,843,836,829,814,721,812,911,1015,1124,1235,1337,1355,1360,1365,1372,1379,1386,1393,1398,1403,1408,1413,1418,1423,1347,1249,1025,1134,1237,1232,1227,1222,1217,1212,1207,1202,1196,1189,1182,1175,1168,1161,1154,1149,1135,1024,1133,1145,921,1026,1040,1047,1054,1061,1068,1075,1082,1088,1093,1098,1103,1108,1113,1118,1123,1127,1130,1132,1244,1344,1426,1484,1479,1474,1469,1464,1457,1450,1443,1438,1433,1429,1427,1422,1335,1233,1122,1013,825,636,723,737,744,751,758,765,772,777,782,787,792,797,802,806,809,811,909,1012,1120,1231,1333,1420,1487,1533,1561,1567,1564,1562,1557,1525,1475,1407,1320,1218,1107,999,897,801,710,558,405,474,488,495,502,509,515,520,525,530,535,539,542,544,624,709,799,895,997,1105,1216,1318,1405,1473,1523,1555,1543,1508,1459,1392,1305,1203,1092,984,882,786,696,612,534,461,345,228,279,293,300,307,312,317,322,326,329,331,393,460,532,610,694,784,880,982,1090,1201,1303,1389,1456,1494,1440,1371,1285,1184,1074,967,866,771,681,597,519,447,381,321,266,186,105,138,152,158,163,167,170,172,216,265,319,379,445,517,595,679,768,863,964,1071,1181,1282,1368,1354,1264,1163,1053,946,845,750,661,578,501,430,365,306,252,204,162,125,81,34,51,62,65,67,93,124,160,202,250,303,362,427,498,575,658,747,842,943,1050,1160,1250,1142,1030,923,822,727,638,555,478,407,342,283,230,183,142,107,78,55,36,23,7,11,16,26,40,59,82,111,146,187,234,287,346,411,482,559,642,731,826,927,1034,1146,1147,1035,928,827,732,643,560,483,412,347,288,235,188,147,112,83,48,18,29,17,28,43,63,88,118,153,194,241,294,353,418,489,566,649,738,833,934,1041,1153,1256,1267,1166,1056,949,848,753,664,581,504,433,368,305,245,191,135,73,98,120,114,99,72,97,130,168,211,259,313,373,439,510,587,670,759,854,955,1062,1172,1273,1361,1432,1374,1288,1187,1077,970,869,770,674,584,500,422,350,276,178,221,261,256,251,244,237,222,177,220,271,327,388,454,526,604,688,778,874,976,1083,1193,1294,1380,1447,1500,1511,1462,1391,1298,1190,1073,959,851,749,653,563,471,337,398,456,451,446,441,435,428,421,414,399,336,397,466,540,619,703,793,889,991,1099,1210,1312,1399,1467,1517,1550,1565,1542,1502,1444,1370,1277,1169,1052,938,830,720,550,629,705,700,695,690,685,680,673,666,659,652,645,630,549,628,715,807,904,1006,1114,1225,1327,1414,1482,1499,1504,1509,1516,1521,1526,1531,1491,1430,1353,1258,1150,1023,817,914,1008,1003,998,993,988,983,978,972,965,958,951,944,937,930,915,816,913,1018,1128,1240,1255,1260,1265,1272,1279,1286,1293,1300,1306,1311,1316,1321,1326,1331,1336,1341,1138,1247,1343,1339,1334,1329,1324,1319,1314,1309,1304,1297,1290,1283,1276,1269,1262,1257,1252,1248,1137,1242,1033,820,919,933,940,947,954,961,968,975,980,985,990,995,1000,1005,1010,1014,1017,1019,1129,1241,1342,1425,1489,1529,1524,1519,1513,1506,1501,1496,1492,1490,1485,1417,1330,1228,1117,1009,906,730,553,634,648,655,662,669,676,682,687,692,697,702,707,711,714,716,808,905,1007,1115,1226,1328,1415,1483,1532,1560,1570,1568,1552,1520,1470,1402,1315,1213,1102,994,892,796,706,621,481,340,403,417,424,431,438,443,448,453,458,462,465,467,541,620,704,794,890,992,1100,1211,1313,1400,1468,1518,1539,1503,1452,1385,1299,1198,1087,979,877,781,691,607,529,457,390,286,181,226,240,247,253,258,263,267,270,272,328,389,455,527,605,689,779,875,977,1085,1195,1296,1382,1449,1435,1364,1278,1177,1067,960,859,764,675,592,514,442,376,316,262,213,145,76,103,117,122,126,129,131,169,212,260,314,374,440,512,589,672,761,856,957,1064,1174,1275,1350,1259,1156,1046,939,838,743,654,571,494,423,358,299,246,199,157,121,90,58,21,32,42,44,64,89,119,155,196,243,296,355,420,491,568,651,740,835,936,1043,1140,1027,920,819,724,635,552,475,404,339,280,227,180,139,104,75,52,33,20,12,6)Polycyclic presentation:
2,3^2,1567|c^a=c^2,d^a=d^1566,d^b=d^1031792 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(28206, 11);