Finite Group DatabasebetaFinite Group Databasebeta| order | 25078 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D12539 |
| unambiguous semidirect product | C12539⋊C2 |
| Fitting, socle, derived, Sylow | C12539 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C12539 | C2 |
| LCS | G ▷ C12539 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12539 |
| transitive degree | μt(G) = 12539 |
A minimal faithful permutation representation — 2 generators on 12539 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1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682)(5860,6681)(5861,6680)(5862,6679)(5863,6678)(5864,6677)(5865,6676)(5866,6675)(5867,6674)(5868,6673)(5869,6672)(5870,6671)(5871,6670)(5872,6669)(5873,6668)(5874,6667)(5875,6666)(5876,6665)(5877,6664)(5878,6663)(5879,6662)(5880,6661)(5881,6660)(5882,6659)(5883,6658)(5884,6657)(5885,6656)(5886,6655)(5887,6654)(5888,6653)(5889,6652)(5890,6651)(5891,6650)(5892,6649)(5893,6648)(5894,6647)(5895,6646)(5896,6645)(5897,6644)(5898,6643)(5899,6642)(5900,6641)(5901,6640)(5902,6639)(5903,6638)(5904,6637)(5905,6636)(5906,6635)(5907,6634)(5908,6633)(5909,6632)(5910,6631)(5911,6630)(5912,6629)(5913,6628)(5914,6627)(5915,6626)(5916,6625)(5917,6624)(5918,6623)(5919,6622)(5920,6621)(5921,6620)(5922,6619)(5923,6618)(5924,6617)(5925,6616)(5926,6615)(5927,6614)(5928,6613)(5929,6612)(5930,6611)(5931,6610)(5932,6609)(5933,6608)(5934,6607)(5935,6606)(5936,6605)(5937,6604)(5938,6603)(5939,6602)(5940,6601)(5941,6600)(5942,6599)(5943,6598)(5944,6597)(5945,6596)(5946,6595)(5947,6594)(5948,6593)(5949,6592)(5950,6591)(5951,6590)(5952,6589)(5953,6588)(5954,6587)(5955,6586)(5956,6585)(5957,6584)(5958,6583)(5959,6582)(5960,6581)(5961,6580)(5962,6579)(5963,6578)(5964,6577)(5965,6576)(5966,6575)(5967,6574)(5968,6573)(5969,6572)(5970,6571)(5971,6570)(5972,6569)(5973,6568)(5974,6567)(5975,6566)(5976,6565)(5977,6564)(5978,6563)(5979,6562)(5980,6561)(5981,6560)(5982,6559)(5983,6558)(5984,6557)(5985,6556)(5986,6555)(5987,6554)(5988,6553)(5989,6552)(5990,6551)(5991,6550)(5992,6549)(5993,6548)(5994,6547)(5995,6546)(5996,6545)(5997,6544)(5998,6543)(5999,6542)(6000,6541)(6001,6540)(6002,6539)(6003,6538)(6004,6537)(6005,6536)(6006,6535)(6007,6534)(6008,6533)(6009,6532)(6010,6531)(6011,6530)(6012,6529)(6013,6528)(6014,6527)(6015,6526)(6016,6525)(6017,6524)(6018,6523)(6019,6522)(6020,6521)(6021,6520)(6022,6519)(6023,6518)(6024,6517)(6025,6516)(6026,6515)(6027,6514)(6028,6513)(6029,6512)(6030,6511)(6031,6510)(6032,6509)(6033,6508)(6034,6507)(6035,6506)(6036,6505)(6037,6504)(6038,6503)(6039,6502)(6040,6501)(6041,6500)(6042,6499)(6043,6498)(6044,6497)(6045,6496)(6046,6495)(6047,6494)(6048,6493)(6049,6492)(6050,6491)(6051,6490)(6052,6489)(6053,6488)(6054,6487)(6055,6486)(6056,6485)(6057,6484)(6058,6483)(6059,6482)(6060,6481)(6061,6480)(6062,6479)(6063,6478)(6064,6477)(6065,6476)(6066,6475)(6067,6474)(6068,6473)(6069,6472)(6070,6471)(6071,6470)(6072,6469)(6073,6468)(6074,6467)(6075,6466)(6076,6465)(6077,6464)(6078,6463)(6079,6462)(6080,6461)(6081,6460)(6082,6459)(6083,6458)(6084,6457)(6085,6456)(6086,6455)(6087,6454)(6088,6453)(6089,6452)(6090,6451)(6091,6450)(6092,6449)(6093,6448)(6094,6447)(6095,6446)(6096,6445)(6097,6444)(6098,6443)(6099,6442)(6100,6441)(6101,6440)(6102,6439)(6103,6438)(6104,6437)(6105,6436)(6106,6435)(6107,6434)(6108,6433)(6109,6432)(6110,6431)(6111,6430)(6112,6429)(6113,6428)(6114,6427)(6115,6426)(6116,6425)(6117,6424)(6118,6423)(6119,6422)(6120,6421)(6121,6420)(6122,6419)(6123,6418)(6124,6417)(6125,6416)(6126,6415)(6127,6414)(6128,6413)(6129,6412)(6130,6411)(6131,6410)(6132,6409)(6133,6408)(6134,6407)(6135,6406)(6136,6405)(6137,6404)(6138,6403)(6139,6402)(6140,6401)(6141,6400)(6142,6399)(6143,6398)(6144,6397)(6145,6396)(6146,6395)(6147,6394)(6148,6393)(6149,6392)(6150,6391)(6151,6390)(6152,6389)(6153,6388)(6154,6387)(6155,6386)(6156,6385)(6157,6384)(6158,6383)(6159,6382)(6160,6381)(6161,6380)(6162,6379)(6163,6378)(6164,6377)(6165,6376)(6166,6375)(6167,6374)(6168,6373)(6169,6372)(6170,6371)(6171,6370)(6172,6369)(6173,6368)(6174,6367)(6175,6366)(6176,6365)(6177,6364)(6178,6363)(6179,6362)(6180,6361)(6181,6360)(6182,6359)(6183,6358)(6184,6357)(6185,6356)(6186,6355)(6187,6354)(6188,6353)(6189,6352)(6190,6351)(6191,6350)(6192,6349)(6193,6348)(6194,6347)(6195,6346)(6196,6345)(6197,6344)(6198,6343)(6199,6342)(6200,6341)(6201,6340)(6202,6339)(6203,6338)(6204,6337)(6205,6336)(6206,6335)(6207,6334)(6208,6333)(6209,6332)(6210,6331)(6211,6330)(6212,6329)(6213,6328)(6214,6327)(6215,6326)(6216,6325)(6217,6324)(6218,6323)(6219,6322)(6220,6321)(6221,6320)(6222,6319)(6223,6318)(6224,6317)(6225,6316)(6226,6315)(6227,6314)(6228,6313)(6229,6312)(6230,6311)(6231,6310)(6232,6309)(6233,6308)(6234,6307)(6235,6306)(6236,6305)(6237,6304)(6238,6303)(6239,6302)(6240,6301)(6241,6300)(6242,6299)(6243,6298)(6244,6297)(6245,6296)(6246,6295)(6247,6294)(6248,6293)(6249,6292)(6250,6291)(6251,6290)(6252,6289)(6253,6288)(6254,6287)(6255,6286)(6256,6285)(6257,6284)(6258,6283)(6259,6282)(6260,6281)(6261,6280)(6262,6279)(6263,6278)(6264,6277)(6265,6276)(6266,6275)(6267,6274)(6268,6273)(6269,6272)(6270,6271)Polycyclic presentation:
2,12539|b^a=b^125386271 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25078, 1);