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D6245 #27159690
Aliases
Splittings
Subgroups
8 conjugacy classes of subgroups.
Full subgroup data →
Presentation
A minimal faithful permutation representation — 2 generators on 1254 points:
(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254);(2,5)(3,4)(7,1254)(8,1253)(9,1252)(10,1251)(11,1250)(12,1249)(13,1248)(14,1247)(15,1246)(16,1245)(17,1244)(18,1243)(19,1242)(20,1241)(21,1240)(22,1239)(23,1238)(24,1237)(25,1236)(26,1235)(27,1234)(28,1233)(29,1232)(30,1231)(31,1230)(32,1229)(33,1228)(34,1227)(35,1226)(36,1225)(37,1224)(38,1223)(39,1222)(40,1221)(41,1220)(42,1219)(43,1218)(44,1217)(45,1216)(46,1215)(47,1214)(48,1213)(49,1212)(50,1211)(51,1210)(52,1209)(53,1208)(54,1207)(55,1206)(56,1205)(57,1204)(58,1203)(59,1202)(60,1201)(61,1200)(62,1199)(63,1198)(64,1197)(65,1196)(66,1195)(67,1194)(68,1193)(69,1192)(70,1191)(71,1190)(72,1189)(73,1188)(74,1187)(75,1186)(76,1185)(77,1184)(78,1183)(79,1182)(80,1181)(81,1180)(82,1179)(83,1178)(84,1177)(85,1176)(86,1175)(87,1174)(88,1173)(89,1172)(90,1171)(91,1170)(92,1169)(93,1168)(94,1167)(95,1166)(96,1165)(97,1164)(98,1163)(99,1162)(100,1161)(101,1160)(102,1159)(103,1158)(104,1157)(105,1156)(106,1155)(107,1154)(108,1153)(109,1152)(110,1151)(111,1150)(112,1149)(113,1148)(114,1147)(115,1146)(116,1145)(117,1144)(118,1143)(119,1142)(120,1141)(121,1140)(122,1139)(123,1138)(124,1137)(125,1136)(126,1135)(127,1134)(128,1133)(129,1132)(130,1131)(131,1130)(132,1129)(133,1128)(134,1127)(135,1126)(136,1125)(137,1124)(138,1123)(139,1122)(140,1121)(141,1120)(142,1119)(143,1118)(144,1117)(145,1116)(146,1115)(147,1114)(148,1113)(149,1112)(150,1111)(151,1110)(152,1109)(153,1108)(154,1107)(155,1106)(156,1105)(157,1104)(158,1103)(159,1102)(160,1101)(161,1100)(162,1099)(163,1098)(164,1097)(165,1096)(166,1095)(167,1094)(168,1093)(169,1092)(170,1091)(171,1090)(172,1089)(173,1088)(174,1087)(175,1086)(176,1085)(177,1084)(178,1083)(179,1082)(180,1081)(181,1080)(182,1079)(183,1078)(184,1077)(185,1076)(186,1075)(187,1074)(188,1073)(189,1072)(190,1071)(191,1070)(192,1069)(193,1068)(194,1067)(195,1066)(196,1065)(197,1064)(198,1063)(199,1062)(200,1061)(201,1060)(202,1059)(203,1058)(204,1057)(205,1056)(206,1055)(207,1054)(208,1053)(209,1052)(210,1051)(211,1050)(212,1049)(213,1048)(214,1047)(215,1046)(216,1045)(217,1044)(218,1043)(219,1042)(220,1041)(221,1040)(222,1039)(223,1038)(224,1037)(225,1036)(226,1035)(227,1034)(228,1033)(229,1032)(230,1031)(231,1030)(232,1029)(233,1028)(234,1027)(235,1026)(236,1025)(237,1024)(238,1023)(239,1022)(240,1021)(241,1020)(242,1019)(243,1018)(244,1017)(245,1016)(246,1015)(247,1014)(248,1013)(249,1012)(250,1011)(251,1010)(252,1009)(253,1008)(254,1007)(255,1006)(256,1005)(257,1004)(258,1003)(259,1002)(260,1001)(261,1000)(262,999)(263,998)(264,997)(265,996)(266,995)(267,994)(268,993)(269,992)(270,991)(271,990)(272,989)(273,988)(274,987)(275,986)(276,985)(277,984)(278,983)(279,982)(280,981)(281,980)(282,979)(283,978)(284,977)(285,976)(286,975)(287,974)(288,973)(289,972)(290,971)(291,970)(292,969)(293,968)(294,967)(295,966)(296,965)(297,964)(298,963)(299,962)(300,961)(301,960)(302,959)(303,958)(304,957)(305,956)(306,955)(307,954)(308,953)(309,952)(310,951)(311,950)(312,949)(313,948)(314,947)(315,946)(316,945)(317,944)(318,943)(319,942)(320,941)(321,940)(322,939)(323,938)(324,937)(325,936)(326,935)(327,934)(328,933)(329,932)(330,931)(331,930)(332,929)(333,928)(334,927)(335,926)(336,925)(337,924)(338,923)(339,922)(340,921)(341,920)(342,919)(343,918)(344,917)(345,916)(346,915)(347,914)(348,913)(349,912)(350,911)(351,910)(352,909)(353,908)(354,907)(355,906)(356,905)(357,904)(358,903)(359,902)(360,901)(361,900)(362,899)(363,898)(364,897)(365,896)(366,895)(367,894)(368,893)(369,892)(370,891)(371,890)(372,889)(373,888)(374,887)(375,886)(376,885)(377,884)(378,883)(379,882)(380,881)(381,880)(382,879)(383,878)(384,877)(385,876)(386,875)(387,874)(388,873)(389,872)(390,871)(391,870)(392,869)(393,868)(394,867)(395,866)(396,865)(397,864)(398,863)(399,862)(400,861)(401,860)(402,859)(403,858)(404,857)(405,856)(406,855)(407,854)(408,853)(409,852)(410,851)(411,850)(412,849)(413,848)(414,847)(415,846)(416,845)(417,844)(418,843)(419,842)(420,841)(421,840)(422,839)(423,838)(424,837)(425,836)(426,835)(427,834)(428,833)(429,832)(430,831)(431,830)(432,829)(433,828)(434,827)(435,826)(436,825)(437,824)(438,823)(439,822)(440,821)(441,820)(442,819)(443,818)(444,817)(445,816)(446,815)(447,814)(448,813)(449,812)(450,811)(451,810)(452,809)(453,808)(454,807)(455,806)(456,805)(457,804)(458,803)(459,802)(460,801)(461,800)(462,799)(463,798)(464,797)(465,796)(466,795)(467,794)(468,793)(469,792)(470,791)(471,790)(472,789)(473,788)(474,787)(475,786)(476,785)(477,784)(478,783)(479,782)(480,781)(481,780)(482,779)(483,778)(484,777)(485,776)(486,775)(487,774)(488,773)(489,772)(490,771)(491,770)(492,769)(493,768)(494,767)(495,766)(496,765)(497,764)(498,763)(499,762)(500,761)(501,760)(502,759)(503,758)(504,757)(505,756)(506,755)(507,754)(508,753)(509,752)(510,751)(511,750)(512,749)(513,748)(514,747)(515,746)(516,745)(517,744)(518,743)(519,742)(520,741)(521,740)(522,739)(523,738)(524,737)(525,736)(526,735)(527,734)(528,733)(529,732)(530,731)(531,730)(532,729)(533,728)(534,727)(535,726)(536,725)(537,724)(538,723)(539,722)(540,721)(541,720)(542,719)(543,718)(544,717)(545,716)(546,715)(547,714)(548,713)(549,712)(550,711)(551,710)(552,709)(553,708)(554,707)(555,706)(556,705)(557,704)(558,703)(559,702)(560,701)(561,700)(562,699)(563,698)(564,697)(565,696)(566,695)(567,694)(568,693)(569,692)(570,691)(571,690)(572,689)(573,688)(574,687)(575,686)(576,685)(577,684)(578,683)(579,682)(580,681)(581,680)(582,679)(583,678)(584,677)(585,676)(586,675)(587,674)(588,673)(589,672)(590,671)(591,670)(592,669)(593,668)(594,667)(595,666)(596,665)(597,664)(598,663)(599,662)(600,661)(601,660)(602,659)(603,658)(604,657)(605,656)(606,655)(607,654)(608,653)(609,652)(610,651)(611,650)(612,649)(613,648)(614,647)(615,646)(616,645)(617,644)(618,643)(619,642)(620,641)(621,640)(622,639)(623,638)(624,637)(625,636)(626,635)(627,634)(628,633)(629,632)(630,631)
Polycyclic presentation:
2,5,1249|b^a=b^4,c^a=c^1248
Conjugacy classes & character table
3124 conjugacy classes. The character table is too large to display.
Relationships
GAP
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12490, 3);
Finite Group Databasebeta