Finite Group DatabasebetaFinite Group Databasebeta| order | 21205 |
| automorphism | ? |
| homology | Abelianization (H1) = C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C4241⋊C5 |
| Fitting, socle, derived, Sylow | C4241 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C4241 | C5 |
| LCS | G ▷ C4241 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 4241 |
| transitive degree | μt(G) = 4241 |
A minimal faithful permutation representation — 2 generators on 4241 points:
(2,3,5,9,17)(4,7,13,25,48)(6,11,21,40,33)(8,15,29,56,105)(10,19,36,18,34)(12,23,44,82,90)(14,27,52,97,64)(16,31,60,113,204)(20,38,71,49,91)(22,42,78,75,65)(24,46,86,155,189)(26,50,93,35,66)(28,54,101,181,163)(30,58,109,196,121)(32,62,117,212,366)(37,69,41,76,68)(39,73,133,106,190)(43,80,144,148,164)(45,84,151,174,122)(47,88,159,275,351)(51,95,170,92,165)(53,99,177,137,123)(55,103,185,317,325)(57,107,192,67,124)(59,111,200,343,283)(61,115,208,358,220)(63,119,216,374,615)(70,128,83,149,167)(72,131,98,175,126)(74,135,238,205,352)(77,139,130,94,168)(79,142,141,138,127)(81,146,253,268,326)(85,153,264,310,284)(87,157,271,336,221)(89,161,279,463,601)(96,172,295,191,327)(100,179,306,257,285)(102,183,313,299,222)(104,187,321,526,575)(108,194,332,166,286)(110,198,339,242,223)(112,202,347,567,534)(114,206,354,125,224)(116,210,362,593,471)(118,214,370,608,382)(120,218,378,623,977)(129,229,156,269,329)(132,233,182,311,288)(134,236,197,337,226)(136,240,407,367,602)(140,245,235,193,330)(143,248,250,258,289)(145,251,261,300,227)(147,255,426,456,576)(150,259,292,169,290)(152,262,303,243,228)(154,266,441,519,535)(158,273,452,560,472)(160,277,459,586,383)(162,281,467,742,963)(171,293,176,301,231)(173,297,489,353,577)(178,304,247,244,232)(180,308,504,445,536)(184,315,515,508,473)(186,319,522,550,384)(188,323,530,831,940)(195,334,546,328,537)(199,341,556,430,474)(201,345,563,493,385)(203,349,571,891,899)(207,356,582,287,475)(209,360,589,411,386)(211,364,597,611,839)(213,368,604,225,387)(215,372,612,957,750)(217,376,619,970,631)(219,380,627,984,1465)(230,393,276,457,579)(234,398,318,520,539)(237,402,344,561,477)(239,405,359,587,389)(241,409,664,616,964)(246,415,404,355,580)(249,418,423,446,540)(252,421,438,509,478)(254,424,449,551,390)(256,428,686,735,941)(260,433,486,331,541)(263,436,501,431,479)(265,439,512,494,391)(267,443,705,824,900)(270,447,543,291,480)(272,450,553,412,392)(274,454,720,884,840)(278,461,731,926,751)(280,465,738,951,632)(282,469,746,1133,1455)(294,484,312,510,482)(296,487,338,410,395)(298,491,775,603,942)(302,496,400,333,544)(305,499,420,432,483)(307,502,435,495,396)(309,506,794,724,901)(314,513,498,413,397)(316,517,809,813,841)(320,524,820,873,752)(322,528,827,915,633)(324,532,835,1250,1432)(335,548,857,578,902)(340,554,417,414,401)(342,558,869,709,842)(346,565,880,798,753)(348,569,887,861,634)(350,573,895,1333,1398)(357,584,911,538,843)(361,591,922,690,754)(363,595,929,779,635)(365,599,936,1122,1340)(369,606,947,476,755)(371,610,954,667,636)(373,614,961,1428,1258)(375,617,966,388,637)(377,621,974,1449,1141)(379,625,980,1459,992)(381,629,988,1472,2067)(394,644,464,736,944)(399,650,527,825,904)(403,655,568,885,845)(406,659,594,927,757)(408,662,609,952,639)(416,672,661,605,945)(419,675,683,725,905)(422,678,702,814,846)(425,681,717,874,758)(427,684,728,916,640)(429,688,1057,1126,1433)(434,694,772,581,906)(437,697,791,710,847)(440,700,806,799,759)(442,703,817,862,641)(444,707,1078,1243,1399)(448,712,854,542,848)(451,715,866,691,760)(453,718,877,780,642)(455,722,1097,1326,1341)(458,726,908,481,761)(460,729,919,668,643)(462,733,1112,956,1259)(466,740,1123,1422,1142)(468,744,1129,1443,993)(470,748,1137,1643,2057)(485,766,521,815,850)(488,770,562,875,763)(490,773,588,917,646)(492,777,1173,965,1434)(497,783,657,583,909)(500,786,680,711,851)(503,789,699,800,764)(505,792,714,666,647)(507,796,1194,1116,1400)(511,801,768,545,852)(514,804,788,692,765)(516,807,803,781,648)(518,811,1213,1232,1342)(523,818,864,669,649)(525,822,1228,1315,1260)(529,829,1239,1377,1143)(531,833,1246,1414,994)(533,837,1254,1784,2044)(547,855,552,689,652)(549,859,1282,943,1401)(555,778,677,693,769)(557,867,696,782,653)(559,871,1296,1101,1343)(564,878,785,670,654)(566,882,1311,1217,1261)(570,889,1322,1300,1144)(572,893,1329,1362,995)(574,897,1337,1891,2012)(585,913,1358,903,1344)(590,920,674,671,658)(592,924,1373,1082,1262)(596,931,1383,1198,1145)(598,934,973,1286,996)(600,938,1394,1774,1968)(607,949,1410,844,1263)(613,959,1425,1175,997)(618,968,1439,756,1146)(620,972,1446,1035,998)(622,976,1453,2040,1792)(624,978,1456,638,999)(626,982,1462,2052,1651)(628,986,1468,2061,1480)(630,990,1476,2074,2727)(645,1007,743,1127,1436)(651,1014,832,1244,1403)(656,1020,892,1327,1346)(660,1025,933,960,1265)(663,1029,958,1423,1148)(665,1032,971,1444,1001)(673,1041,1031,967,1437)(676,1043,1054,1117,1404)(679,1046,1075,1233,1347)(682,1049,1094,1316,1266)(685,1052,1109,1378,1149)(687,1055,1120,1415,1002)(695,1064,1170,946,1405)(698,1067,1191,1102,1348)(701,1070,1210,1218,1267)(704,1073,1225,1301,1150)(706,1076,1236,1363,1003)(708,1080,1571,1778,2013)(713,1086,1280,907,1349)(716,1089,1293,1083,1268)(719,1092,1308,1199,1151)(721,1095,1319,1287,1004)(723,1099,1592,1884,1969)(727,1104,1355,849,1269)(730,1107,1370,1060,1152)(732,1110,1380,1176,1005)(734,1114,1611,1621,1897)(737,1118,1407,762,1153)(739,1121,1418,1036,1006)(741,1125,1625,2006,1793)(745,1131,1634,2035,1652)(747,1135,1639,2049,1481)(749,1139,1647,2245,2717)(767,1159,826,1234,1351)(771,1164,886,1317,1271)(774,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666,3517,1095,101,182,312,511,802,1203,1717,1864,311,509,799,1199,1712,1517,2117,2768,3369,3820,4092,4209,4178,2425,2204,2835,3411,2694,3304,3777,4070,3220,3117,820,1225,1743,2340,2952,2789,372,613,960,1427,2005,2645,3257,1457,1164,1674,2274,2903,629,989,1475,2073,1907,2535,3155,3665,4003,1940,2572,1135,1640,2236,2865,3435,2727,3336,2713,2306,405,658,1023,1514,2114,2765,3366,3183,852,1273,1808,2422,3041,3568,2402,3021,1401,1971,2606,3218,3709,1600,2190,1057,1548,2143,2788,3383,3829,4094,2241,1835,295,486,768,1161,119,217,377,622,580,905,1347,1904,2532,591,923,1372,1934,2565,3181,3684,3636,3979,3082,3595,3952,4143,3386,707,1079,1570,2162,1373,1935,2566,3182,1694,1430,2010,2651,2032,366,601,940,1398,1968,457,725,1102,1597,2187,1766,2369,2982,3518,3903,4122,4216,4220,2934,675,15,30,59,112,203,350,574,898,1123,1622,2216,1459,2049,2697,3308,3780,1885,2508,3124,3633,3976,3890,1563,153,265,440,701,1071,1561,2153,2460,496,782,1178,1692,2293,2918,3475,3880,3142,3652,3995,3331,3796,4079,2817,528,828,1238,1760,2361,2974,3511,3606,1208,142)Polycyclic presentation:
5,4241|b^a=b^1395853 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21205, 1);