Finite Group DatabasebetaFinite Group Databasebeta| order | 50653 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C372 · Schur multiplier (H2) = C372 |
| unitriangular | UT(3,𝔽37) |
| extraspecial | 37+1+2 |
| Heisenberg | H(3,𝔽37) |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C37 | C372 |
| LCS | G ▷ C37 ▷ C1 | |
| UCS | C1 ◁ C37 ◁ G | |
| LECS | G ▷ C37 ▷ C1 | |
79 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1369 |
| transitive degree | μt(G) = 1369 |
A minimal faithful permutation representation — 2 generators on 1369 points:
(1,2,4,8,15,26,42,64,93,130,176,232,299,358,407,456,499,542,585,628,671,710,749,788,827,866,905,944,983,1022,1061,1100,1139,1178,1217,1256,1295)(3,6,10,17,28,44,66,95,132,178,234,301,360,409,458,501,544,587,630,673,712,751,790,829,868,907,946,985,1024,1063,1102,1141,1180,1219,1258,1297,1334)(5,11,21,37,59,88,125,171,227,294,353,402,451,495,538,581,624,667,708,747,786,825,864,903,942,981,1020,1059,1098,1137,1176,1215,1254,1293,1332,1369,1333)(7,13,20,31,47,69,98,135,181,237,304,363,412,461,504,547,590,633,675,714,753,792,831,870,909,948,987,1026,1065,1104,1143,1182,1221,1260,1299,1296,1335)(9,18,32,53,83,120,166,222,289,348,399,448,494,537,580,623,666,707,746,785,824,863,902,941,980,1019,1058,1097,1136,1175,1214,1253,1292,1331,1368,1294,12)(14,24,36,52,74,103,140,186,242,309,368,417,465,508,551,594,637,632,677,717,757,797,837,877,917,957,997,1037,1077,1117,1157,1197,1237,1277,1257,1298,1336)(16,29,48,75,112,160,216,283,345,398,447,493,536,579,622,665,706,745,784,823,862,901,940,979,1018,1057,1096,1135,1174,1213,1252,1291,1330,1367,1255,23,22)(19,34,27,45,70,104,149,207,276,343,397,446,492,535,578,621,664,705,744,783,822,861,900,939,978,1017,1056,1095,1134,1173,1212,1251,1290,1329,1366,1216,39)(25,40,58,82,111,148,194,250,317,375,424,416,469,515,561,607,653,635,679,719,759,799,839,879,919,959,999,1039,1079,1119,1159,1199,1239,1218,1259,1300,1337)(30,50,78,65,96,136,187,251,329,393,443,490,533,576,619,662,703,742,781,820,859,898,937,976,1015,1054,1093,1132,1171,1210,1249,1288,1327,1364,1138,90,35)(33,54,43,67,99,141,195,263,340,396,445,491,534,577,620,663,704,743,782,821,860,899,938,977,1016,1055,1094,1133,1172,1211,1250,1289,1328,1365,1177,61,38)(41,62,87,119,159,206,262,328,316,382,437,420,472,518,564,610,589,636,680,720,760,800,840,880,920,960,1000,1040,1080,1120,1160,1200,1179,1220,1261,1301,1338)(46,72,107,153,131,179,238,310,376,431,482,528,572,616,660,701,740,779,818,857,896,935,974,1013,1052,1091,1130,1169,1208,1247,1286,1325,1362,1060,173,57,55)(49,76,113,94,133,182,243,318,383,438,487,531,575,618,661,702,741,780,819,858,897,936,975,1014,1053,1092,1131,1170,1209,1248,1287,1326,1363,1099,127,60,56)(51,80,71,105,150,208,177,235,305,369,425,476,522,568,613,657,699,739,778,817,856,895,934,973,1012,1051,1090,1129,1168,1207,1246,1285,1324,1361,1021,229,89)(63,91,124,165,215,275,339,337,321,386,441,422,474,520,566,612,592,639,682,722,762,802,842,882,922,962,1002,1042,1082,1122,1162,1140,1181,1222,1262,1302,1339)(68,101,144,199,268,233,302,364,418,470,516,562,608,654,696,736,776,816,855,894,933,972,1011,1050,1089,1128,1167,1206,1245,1284,1323,1360,982,296,86,85,79)(73,109,97,138,190,255,333,395,359,410,462,509,555,601,647,690,730,770,810,850,890,930,970,1009,1048,1087,1126,1165,1204,1243,1282,1321,1358,904,404,123,81)(77,114,100,142,196,264,341,300,361,413,466,512,558,604,650,693,733,773,813,853,893,932,971,1010,1049,1088,1127,1166,1205,1244,1283,1322,1359,943,355,126,84)(92,128,170,221,282,274,261,338,324,389,367,423,475,521,567,546,593,640,683,723,763,803,843,883,923,963,1003,1043,1083,1123,1101,1142,1183,1223,1263,1303,1340)(102,146,202,183,244,319,384,439,488,532,500,545,591,638,681,721,761,801,841,881,921,961,1001,1041,1081,1121,1161,1201,1240,1279,1318,1355,787,539,228,118,115)(106,151,134,184,246,322,387,442,489,457,502,548,595,641,684,724,764,804,844,884,924,964,1004,1044,1084,1124,1163,1202,1241,1280,1319,1356,826,496,169,121,116)(108,154,137,188,252,330,394,444,408,459,505,552,598,644,687,727,767,807,847,887,927,967,1007,1047,1086,1125,1164,1203,1242,1281,1320,1357,865,453,172,122,117)(110,157,145,200,180,240,313,379,434,485,530,574,543,588,634,678,718,758,798,838,878,918,958,998,1038,1078,1118,1158,1198,1238,1278,1317,1354,748,582,225,168)(129,174,226,288,344,277,267,342,326,391,371,427,478,524,570,549,596,642,685,725,765,805,845,885,925,965,1005,1045,1085,1062,1103,1144,1184,1224,1264,1304,1341)(139,192,258,236,307,372,428,479,525,571,615,659,629,674,715,755,795,835,875,915,955,995,1035,1075,1115,1155,1195,1235,1275,1315,1352,670,668,292,164,162,155)(143,197,265,239,311,377,432,483,529,573,617,586,631,676,716,756,796,836,876,916,956,996,1036,1076,1116,1156,1196,1236,1276,1316,1353,709,625,295,167,163,156)(147,204,189,253,331,303,366,421,473,519,565,611,656,698,738,711,752,793,833,873,913,953,993,1033,1073,1113,1153,1193,1233,1273,1313,1350,584,626,351,223,158)(152,209,191,256,334,306,370,426,477,523,569,614,658,700,672,713,754,794,834,874,914,954,994,1034,1074,1114,1154,1194,1234,1274,1314,1351,627,669,354,224,161)(175,230,293,347,281,278,270,249,327,392,373,429,480,526,503,550,597,643,686,726,766,806,846,886,926,966,1006,1046,1023,1064,1105,1145,1185,1225,1265,1305,1342)(185,248,325,390,365,419,471,517,563,609,655,697,737,777,750,791,832,872,912,952,992,1032,1072,1112,1152,1192,1232,1272,1312,1349,541,583,403,220,219,213,203)(193,260,241,315,381,436,411,464,511,557,603,649,692,732,772,812,852,892,867,908,949,989,1029,1069,1109,1149,1189,1229,1269,1309,1346,406,454,449,287,214,210)(198,266,245,320,385,440,414,467,513,559,605,651,694,734,774,814,854,828,869,910,950,990,1030,1070,1110,1150,1190,1230,1270,1310,1347,455,497,452,290,217,211)(201,269,247,323,388,362,415,468,514,560,606,652,695,735,775,815,789,830,871,911,951,991,1031,1071,1111,1151,1191,1231,1271,1311,1348,498,540,400,291,218,212)(205,273,259,336,314,380,435,486,463,510,556,602,648,691,731,771,811,851,891,931,906,947,988,1028,1068,1108,1148,1188,1228,1268,1308,1345,357,405,450,350,286)(231,297,352,346,284,279,271,254,332,308,374,430,481,527,506,553,599,645,688,728,768,808,848,888,928,968,1008,984,1025,1066,1106,1146,1186,1226,1266,1306,1343)(257,335,312,378,433,484,460,507,554,600,646,689,729,769,809,849,889,929,969,945,986,1027,1067,1107,1147,1187,1227,1267,1307,1344,298,356,401,349,285,280,272);(2,5,12,23,39,61,90,127,173,229,296,355,404,453,496,539,582,625,668,669,626,583,540,497,454,405,356,297,230,174,128,91,62,40,24,13,6)(4,9,19,35,57,86,123,169,225,292,351,400,449,401,293,170,87,36,10,11,22,38,60,89,126,172,228,295,354,403,452,450,352,226,124,58,20)(8,16,30,51,81,118,164,220,287,346,221,82,17,18,33,55,84,121,167,223,290,349,288,119,31,21,34,56,85,122,168,224,291,350,347,165,52)(15,27,46,73,110,158,214,281,159,28,29,49,77,115,161,217,284,215,47,32,50,79,116,162,218,285,282,74,37,54,80,117,163,219,286,344,111)(26,43,68,102,147,205,274,69,48,72,108,155,211,278,148,59,78,114,157,213,280,275,44,45,71,106,152,210,277,103,53,76,109,156,212,279,206)(42,65,97,139,193,261,66,67,100,143,198,267,98,70,101,145,201,270,140,75,105,146,203,271,194,83,107,151,204,272,262,88,113,154,209,273,339)(64,94,134,185,249,135,99,138,191,257,250,112,144,197,260,337,125,153,202,269,342,95,96,137,189,254,186,104,142,192,259,328,120,150,200,266,338)(93,131,180,241,316,160,196,256,332,181,136,184,247,324,171,208,265,336,317,149,190,253,327,132,133,183,245,321,166,199,258,335,242,141,188,248,326)(130,177,236,308,178,179,239,312,237,182,240,314,309,187,244,315,375,195,246,320,382,207,252,323,386,216,255,325,389,222,264,331,391,227,268,334,392)(176,233,303,367,283,330,385,424,251,313,378,234,235,306,371,289,333,388,437,263,319,380,304,238,307,373,294,341,390,441,276,322,381,368,243,311,374)(232,300,362,416,318,372,427,345,387,435,301,302,365,420,329,377,429,348,394,436,363,305,366,422,340,379,430,353,395,440,417,310,370,423,343,384,433)(299,359,411,360,361,414,412,364,415,465,369,419,469,376,421,472,383,426,474,393,428,475,396,432,478,397,434,480,398,439,481,399,442,484,402,444,486)(358,408,460,448,488,526,446,483,521,443,477,518,431,471,508,418,467,409,410,463,451,489,527,447,485,524,445,479,520,438,473,515,425,468,461,413,464)(407,457,503,491,523,561,470,511,495,532,570,490,519,551,466,510,494,530,567,487,517,504,462,507,493,529,566,482,514,458,459,506,492,525,564,476,513)(456,500,546,528,559,538,574,612,522,557,537,573,610,516,556,536,571,607,512,554,535,569,594,509,553,534,565,547,505,550,533,563,501,502,549,531,560)(499,543,589,558,599,576,606,581,617,653,555,597,575,605,580,615,637,552,596,572,603,579,614,590,548,593,568,602,578,611,544,545,592,562,600,577,609)(542,586,632,595,639,604,643,616,648,620,652,623,658,587,588,635,598,640,608,645,618,649,621,655,624,659,633,591,636,601,642,613,646,619,651,622,656)(585,629,630,631,675,634,677,638,679,641,680,644,682,647,683,650,685,654,686,657,688,660,689,661,691,662,692,663,694,664,695,665,697,666,698,667,700)(628,672,708,738,707,737,706,735,705,734,704,732,703,731,702,729,701,728,699,726,696,725,693,723,690,722,687,720,684,719,681,717,678,714,676,673,674)(671,711,745,774,742,769,739,765,730,760,721,753,715,747,777,744,772,741,768,736,763,727,759,718,712,713,746,775,743,771,740,766,733,762,724,757,716)(710,750,782,809,776,802,761,751,752,783,811,778,803,764,792,754,784,812,779,805,767,797,755,785,814,780,806,770,799,756,786,815,781,808,773,800,758)(749,789,819,845,804,790,791,820,846,807,831,793,821,848,810,837,794,822,849,813,839,795,823,851,816,840,796,824,852,817,842,798,825,854,818,843,801)(788,828,856,880,835,861,888,847,829,830,857,882,836,862,889,850,870,832,858,883,838,863,891,853,877,833,859,885,841,864,892,855,879,834,860,886,844)(827,867,893,909,871,895,919,873,897,922,875,899,925,878,901,928,884,903,931,890,868,869,894,917,872,896,920,874,898,923,876,900,926,881,902,929,887)(866,906,927,942,969,924,941,968,921,940,966,918,939,965,916,938,963,915,937,962,914,936,960,913,935,959,912,934,957,911,933,948,910,932,907,908,930)(905,945,961,978,1003,954,974,997,950,970,981,1008,958,977,1002,953,973,987,949,967,980,1006,956,976,1000,952,972,946,947,964,979,1005,955,975,999,951,971)(944,984,996,1014,1037,989,1004,1017,1042,992,1010,1020,1046,995,1013,1026,988,1001,1016,1040,991,1009,1019,1045,994,1012,985,986,998,1015,1039,990,1007,1018,1043,993,1011)(983,1023,1034,1050,1059,1085,1033,1049,1058,1083,1032,1048,1057,1082,1031,1047,1056,1080,1030,1044,1055,1079,1029,1041,1054,1077,1028,1038,1053,1065,1027,1036,1052,1024,1025,1035,1051)(1022,1062,1072,1086,1094,1117,1067,1075,1089,1097,1122,1070,1081,1092,1063,1064,1073,1087,1095,1119,1068,1076,1090,1098,1123,1071,1084,1093,1104,1066,1074,1088,1096,1120,1069,1078,1091)(1061,1101,1110,1118,1129,1136,1160,1108,1115,1127,1134,1157,1106,1113,1125,1132,1102,1103,1111,1121,1130,1137,1162,1109,1116,1128,1135,1159,1107,1114,1126,1133,1143,1105,1112,1124,1131)(1100,1140,1148,1154,1164,1170,1176,1200,1147,1153,1163,1169,1175,1199,1146,1152,1161,1168,1174,1197,1145,1151,1158,1167,1173,1182,1144,1150,1156,1166,1172,1141,1142,1149,1155,1165,1171)(1139,1179,1186,1191,1196,1204,1209,1214,1237,1184,1189,1194,1202,1207,1212,1180,1181,1187,1192,1198,1205,1210,1215,1239,1185,1190,1195,1203,1208,1213,1221,1183,1188,1193,1201,1206,1211)(1178,1218,1224,1228,1232,1236,1242,1246,1250,1254,1277,1223,1227,1231,1235,1241,1245,1249,1253,1260,1222,1226,1230,1234,1240,1244,1248,1252,1219,1220,1225,1229,1233,1238,1243,1247,1251)(1217,1257,1262,1265,1268,1271,1274,1278,1281,1284,1287,1290,1293,1299,1261,1264,1267,1270,1273,1276,1280,1283,1286,1289,1292,1258,1259,1263,1266,1269,1272,1275,1279,1282,1285,1288,1291)(1256,1296,1300,1302,1304,1306,1308,1310,1312,1314,1316,1318,1320,1322,1324,1326,1328,1330,1332,1297,1298,1301,1303,1305,1307,1309,1311,1313,1315,1317,1319,1321,1323,1325,1327,1329,1331)(1295,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369)Polycyclic presentation:
37^3|b^a=bc1405 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(50653, 3);