Finite Group DatabasebetaFinite Group Databasebeta| order | 22405 |
| automorphism | ? |
| homology | Abelianization (H1) = C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C4481⋊C5 |
| Fitting, socle, derived, Sylow | C4481 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C4481 | C5 |
| LCS | G ▷ C4481 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 4481 |
| transitive degree | μt(G) = 4481 |
A minimal faithful permutation representation — 2 generators on 4481 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)(63,119,216,374,617)(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)(89,161,279,463,602)(96,172,295,191,327)(100,179,306,257,285)(102,183,313,299,222)(104,187,321,526,576)(108,194,332,166,286)(110,198,339,242,223)(112,202,347,568,534)(114,206,354,125,224)(116,210,362,594,471)(118,214,370,609,382)(120,218,378,625,984)(129,229,156,269,329)(132,233,182,311,288)(134,236,197,337,226)(136,240,407,367,603)(140,245,235,193,330)(143,248,250,258,289)(145,251,261,300,227)(147,255,426,456,577)(150,259,292,169,290)(152,262,303,243,228)(154,266,441,519,535)(158,273,452,561,472)(160,277,459,587,383)(162,281,467,745,969)(171,293,176,301,231)(173,297,489,353,578)(178,304,247,244,232)(180,308,504,445,536)(184,315,515,508,473)(186,319,522,550,384)(188,323,530,833,943)(195,334,546,328,537)(199,341,557,430,474)(201,345,564,493,385)(203,349,572,895,903)(207,356,583,287,475)(209,360,590,411,386)(211,364,598,935,841)(213,368,605,225,387)(215,372,613,961,753)(217,376,621,976,633)(219,380,629,991,1486)(230,393,276,457,580)(234,398,318,520,539)(237,402,344,562,477)(239,405,359,588,389)(241,409,666,618,970)(246,415,404,355,581)(249,418,423,446,540)(252,421,438,509,478)(254,424,449,551,390)(256,428,689,738,944)(260,433,486,331,541)(263,436,501,431,479)(265,439,512,494,391)(267,443,708,826,904)(270,447,543,291,480)(272,450,554,412,392)(274,454,723,888,842)(278,461,734,928,754)(280,465,741,954,634)(282,469,749,1146,1474)(294,484,312,510,482)(296,487,338,552,395)(298,491,778,604,945)(302,496,400,333,544)(305,499,420,432,483)(307,502,435,495,396)(309,506,797,727,905)(314,513,498,413,397)(316,517,812,816,843)(320,524,628,877,755)(322,528,829,917,635)(324,532,837,1261,1448)(335,548,653,579,834)(340,555,417,414,401)(342,559,873,712,844)(346,566,884,801,756)(348,570,891,862,636)(350,574,899,1345,1409)(357,585,914,538,845)(361,592,924,693,757)(363,596,931,782,637)(365,600,939,1401,1353)(369,607,950,476,758)(371,611,957,670,638)(373,615,965,1440,1269)(375,619,972,388,639)(377,623,980,1466,1154)(379,627,988,1481,999)(381,631,995,1492,2121)(394,646,464,739,947)(399,652,527,827,907)(403,657,569,889,847)(406,661,595,929,760)(408,664,610,955,641)(410,668,1040,985,1475)(416,675,663,606,948)(419,678,686,728,908)(422,681,705,817,848)(425,684,720,878,761)(427,687,731,918,642)(429,691,1067,1139,1449)(434,697,775,582,909)(437,700,794,713,849)(440,703,809,802,762)(442,706,820,863,643)(444,710,1090,1255,1410)(448,715,856,542,850)(451,718,870,694,763)(453,721,881,783,644)(455,725,1109,1339,1354)(458,729,911,481,764)(460,732,921,671,645)(462,736,1124,1394,1270)(466,743,1135,1433,1155)(468,747,1142,1459,1000)(470,751,1150,1679,2111)(485,769,521,818,852)(488,773,563,879,766)(490,776,589,919,648)(492,780,1187,971,1450)(497,786,659,584,912)(500,789,683,714,853)(503,792,702,803,767)(505,795,717,864,649)(507,799,1209,1128,1411)(511,804,771,545,854)(514,807,791,695,768)(516,810,806,784,650)(518,814,1228,1244,1355)(523,821,867,672,651)(525,824,1149,1328,1271)(529,831,1251,1383,1156)(531,835,1257,1422,1001)(533,839,1265,1822,2093)(547,857,553,865,654)(549,860,1057,946,1412)(556,868,680,696,772)(558,871,699,785,655)(560,875,1174,1113,1256)(565,882,788,673,656)(567,886,1324,1232,1272)(571,893,1335,1313,1157)(573,897,1342,1370,1002)(575,901,1349,1924,2055)(586,916,1368,906,1258)(591,922,677,674,660)(593,926,1380,1094,1273)(597,933,1390,1213,1158)(599,937,1397,1295,1003)(601,941,1405,1996,2003)(608,952,1419,846,1274)(612,959,1429,1071,1159)(614,963,1436,1190,1004)(616,967,1444,2047,1932)(620,974,1455,759,1160)(622,978,1462,1044,1005)(624,982,1470,2085,1828)(626,986,1477,640,1006)(630,993,1490,2117,1500)(632,997,1496,2127,2843)(647,1014,746,1140,1452)(658,1026,896,1340,1357)(662,1031,936,1395,1276)(665,1035,962,1434,1162)(667,1038,977,1460,1008)(669,1042,1548,1487,2112)(676,1050,1037,973,1453)(679,1053,1064,1129,1413)(682,1056,1087,1245,1358)(685,1059,1106,1329,1277)(688,1062,1121,1384,1163)(690,1065,1132,1423,1009)(692,1069,1499,1673,2094)(698,1076,1184,949,1414)(701,1079,1206,1114,1359)(704,1082,1225,1233,1278)(707,1085,990,1314,1164)(709,1088,1248,1371,1010)(711,1092,1602,1816,2056)(716,1098,1021,910,1262)(719,1101,1307,1095,1279)(722,1104,1321,1214,1165)(724,1107,1332,1296,1011)(726,1111,1625,1919,2004)(730,1116,1365,851,1280)(733,1119,1377,1072,1166)(735,1122,1387,1191,1012)(737,1126,1643,1990,1933)(740,1130,1416,765,1167)(742,1133,1426,1045,1013)(744,1137,1658,2040,1829)(748,1144,1669,2078,1687)(750,1148,1676,2103,1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presentation:
5,4481|b^a=b^475901 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22405, 1);