Finite Group DatabasebetaFinite Group Databasebeta| order | 14641 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C112 · Schur multiplier (H2) = C11 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C11 | H(3,𝔽11) |
| commutator, Frattini, cosocle | C112 | C112 |
| prime radical | H(3,𝔽11) | C11 |
| LCS | G ▷ C112 ▷ C11 ▷ C1 | |
| UCS | C1 ◁ C11 ◁ C112 ◁ G | |
| LECS | G ▷ C112 ▷ C11 ▷ C1 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1331 |
| transitive degree | μt(G) = 1331 |
A minimal faithful permutation representation — 2 generators on 1331 points:
(1,2,5,13,31,68,136,225,317,419,532)(3,9,23,53,111,198,293,395,508,632,656)(4,8,16,34,71,139,228,320,422,535,657)(6,17,41,90,172,268,373,489,533,658,11)(7,20,49,106,194,292,394,507,631,765,655)(10,26,59,121,173,269,374,490,616,681,787)(12,22,40,77,144,233,324,427,539,534,661)(14,35,78,155,253,355,420,536,662,66,48)(15,38,85,167,266,372,488,614,751,531,21)(18,45,99,156,256,356,471,548,659,790,65)(19,44,93,147,244,345,451,549,660,789,29)(24,56,73,149,245,346,424,544,675,788,28)(25,43,97,69,140,226,321,428,547,680,786)(27,62,126,211,246,348,460,551,685,820,900)(30,52,89,154,143,238,337,452,421,538,667)(32,72,145,239,318,423,540,668,315,291,166)(33,75,151,250,353,470,599,738,418,51,50)(36,81,160,259,362,476,537,665,796,416,290)(37,80,157,137,229,325,433,557,692,134,105)(39,87,70,142,237,336,450,580,720,316,109)(42,94,178,274,378,492,615,663,793,222,104)(46,101,186,275,341,455,567,704,791,903,221)(47,54,113,200,234,326,437,562,697,792,133)(55,92,177,255,327,439,542,674,808,654,58)(57,118,175,272,333,438,564,695,809,902,132)(60,123,161,257,330,442,569,673,806,901,64)(61,117,159,199,240,340,434,560,694,815,899)(63,129,214,279,360,445,570,696,819,918,996)(67,110,171,170,152,251,354,319,426,546,679)(74,148,241,339,425,543,671,800,417,207,115)(76,153,252,227,323,432,555,691,135,197,88)(79,141,235,322,430,541,672,804,785,125,122)(82,162,262,342,454,563,666,798,911,898,213)(83,91,174,254,358,472,558,669,801,529,301)(84,107,195,231,328,436,559,664,795,223,193)(86,169,138,232,331,444,573,712,224,196,108)(95,150,248,332,441,545,677,797,909,652,393)(96,180,146,242,338,429,550,683,817,314,192)(98,112,168,267,357,435,556,670,803,530,120)(100,184,260,347,461,561,682,794,906,415,191)(102,188,263,364,463,589,699,807,904,1000,414)(103,116,158,243,343,453,552,676,810,905,313)(114,201,271,359,473,600,698,818,908,528,289)(119,205,278,344,458,581,688,821,921,999,312)(124,208,249,351,456,583,678,813,919,998,220)(127,163,264,368,447,565,701,799,913,997,131)(128,164,179,270,363,477,603,693,805,917,995)(130,217,280,379,482,577,702,834,929,1010,1080)(165,176,247,230,329,440,566,703,836,653,210)(181,258,349,464,588,687,812,912,1009,1079,216)(182,183,236,334,431,553,684,802,915,784,303)(185,273,366,457,582,700,816,910,1006,897,404)(187,282,350,446,568,686,822,907,1003,651,288)(189,285,361,466,576,706,824,926,1001,1085,650)(190,204,276,376,443,571,708,829,922,1002,527)(202,281,375,462,590,705,830,916,1008,994,305)(203,261,335,448,554,689,811,923,1005,783,392)(206,265,370,484,574,709,837,914,1012,1084,526)(209,299,377,478,592,728,832,930,1016,1083,413)(212,277,365,459,586,721,826,925,1011,1082,311)(215,294,384,467,584,723,823,933,1015,1081,219)(218,308,398,475,595,724,854,935,1023,1099,1152)(283,381,465,572,710,825,932,1007,1091,1078,406)(284,369,480,585,722,833,920,1004,1088,896,391)(286,388,493,606,725,853,943,1026,1086,1158,895)(287,297,382,449,578,690,827,931,1018,1087,782)(295,367,474,575,707,831,924,1014,1093,1151,307)(296,352,468,591,729,814,927,1019,1090,993,506)(298,380,481,579,718,828,939,1021,1096,1157,781)(300,396,495,594,716,838,936,1029,1101,1156,649)(302,383,469,597,714,840,928,1024,1095,1155,525)(304,397,483,604,711,842,942,1020,1100,1154,412)(306,385,496,587,726,852,938,1028,1094,1153,310)(309,409,500,607,727,857,962,1035,1106,1165,1212)(371,486,596,730,859,934,1013,1097,1160,992,390)(386,485,608,717,835,944,1017,1092,1164,1211,408)(387,479,593,713,841,937,1022,1089,1161,1077,505)(389,502,620,731,845,949,1034,1113,1159,1219,1076)(399,491,601,715,839,941,1027,1103,1163,1150,518)(400,497,602,734,843,951,1031,1108,1168,1218,991)(401,509,610,743,856,963,1040,1104,1174,1217,894)(402,498,617,719,850,940,1036,1112,1167,1216,780)(403,510,605,732,848,946,1033,1107,1173,1215,648)(405,499,611,735,846,948,1030,1102,1166,1214,524)(407,511,618,741,844,950,1039,1111,1172,1213,411)(410,521,619,746,870,954,1046,1124,1177,1230,1260)(487,612,733,855,964,1032,1098,1170,1224,1259,520)(494,609,739,847,947,1038,1105,1169,1221,1149,504)(501,598,736,849,945,1025,1109,1162,1222,1210,630)(503,627,747,865,959,1041,1114,1176,1220,1268,1209)(512,622,753,868,957,1043,1123,1184,1228,1267,1148)(513,633,744,860,955,1045,1119,1179,1234,1266,1075)(514,623,748,863,953,1047,1116,1175,1227,1265,990)(515,634,737,866,966,1060,1110,1182,1233,1264,893)(516,624,745,851,960,1037,1121,1178,1226,1263,779)(517,613,749,861,958,1042,1118,1171,1231,1262,647)(519,625,740,864,956,1044,1115,1181,1225,1261,523)(522,644,761,876,972,1054,1128,1188,1236,1272,1296)(621,742,862,965,1061,1117,1180,1229,1270,1258,629)(626,752,858,952,1048,1120,1185,1223,1271,1295,643)(628,762,873,967,1050,1132,1193,1239,1269,1305,1294)(635,757,877,970,1062,1145,1190,1235,1276,1304,1257)(636,766,869,973,1057,1125,1187,1242,1282,1303,1208)(637,758,874,977,1055,1127,1196,1238,1275,1302,1147)(638,750,878,968,1053,1129,1192,1232,1280,1301,1074)(639,759,871,971,1051,1131,1189,1241,1274,1300,989)(640,754,875,974,1049,1133,1186,1237,1279,1299,892)(641,760,880,961,1058,1122,1194,1245,1273,1298,778)(642,755,872,969,1056,1126,1191,1240,1278,1297,646)(645,776,884,986,1065,1142,1198,1252,1283,1313,1320)(756,867,975,1052,1130,1183,1243,1277,1307,1319,764)(763,890,979,1068,1140,1200,1249,1290,1306,1330,1331)(767,885,983,1071,1138,1202,1246,1286,1312,1329,1318)(768,879,987,1063,1136,1204,1254,1281,1317,1328,1293)(769,886,980,1066,1134,1206,1251,1289,1311,1327,1256)(770,881,984,1069,1143,1197,1248,1285,1316,1326,1207)(771,887,976,1072,1141,1199,1244,1292,1310,1325,1146)(772,882,981,1064,1139,1201,1253,1288,1315,1324,1073)(773,888,985,1067,1137,1203,1250,1284,1309,1323,988)(774,883,978,1070,1135,1205,1247,1291,1314,1322,891)(775,889,982,1059,1144,1195,1255,1287,1308,1321,777);(1,3,10,27,63,130,218,309,410,522,645,12,29,65,132,220,311,412,524,647,778,891,67,134,222,313,414,526,649,780,893,989,1073,224,315,416,528,651,782,895,991,1075,1147,1207,418,530,653,784,897,993,1077,1149,1209,1257,1293,655,786,899,995,1079,1151,1211,1259,1295,1319,1331,4,11,28,64,131,219,310,411,523,646,777,30,66,133,221,312,413,525,648,779,892,988,135,223,314,415,527,650,781,894,990,1074,1146,316,417,529,652,783,896,992,1076,1148,1208,1256,531,654,785,898,994,1078,1150,1210,1258,1294,1318)(2,6,18,46,102,189,286,389,503,628,763,22,48,104,191,288,391,505,630,643,644,775,110,193,290,393,404,406,408,409,519,641,773,196,207,210,213,216,217,306,405,516,639,771,51,58,61,62,127,212,302,402,514,637,769,7,9,24,57,119,206,298,400,512,635,767,8,19,47,103,190,287,390,504,629,764,776,52,105,192,289,392,506,518,520,521,642,774,197,291,301,303,305,307,308,407,517,640,772,109,120,125,128,129,215,304,403,515,638,770,21,25,26,60,124,209,300,401,513,636,768)(5,14,36,82,163,265,371,487,613,750,879,40,84,165,126,205,297,399,511,634,766,890,171,115,117,118,204,296,398,510,633,762,889,108,55,56,116,203,295,397,509,627,761,888,50,23,54,114,202,294,396,502,626,760,887,20,44,96,182,214,299,388,501,625,759,886,16,37,83,164,208,285,387,500,624,758,885,89,166,122,123,188,284,386,499,623,757,884,88,98,59,101,187,283,385,498,622,756,883,39,43,45,100,185,280,383,497,621,755,882,15,17,42,95,181,277,380,494,619,754,881)(13,32,73,150,249,352,469,598,737,867,976,77,112,113,162,263,367,481,607,744,872,980,170,97,180,211,282,384,486,611,747,875,983,86,41,91,175,273,377,491,617,752,878,986,33,35,79,158,258,361,475,602,740,869,978,49,107,121,201,264,369,483,609,745,873,981,34,74,99,183,278,381,495,612,748,876,984,154,92,94,179,276,379,493,618,753,880,987,76,53,81,161,261,365,479,605,742,871,979,87,93,176,186,281,370,485,610,746,874,982,38,80,159,184,279,382,496,620,749,877,985)(31,69,141,236,335,449,579,719,851,961,1059,144,90,173,270,375,468,596,734,863,971,1067,152,78,156,257,360,474,601,739,868,977,1072,169,195,178,275,351,467,595,733,862,970,1066,75,148,160,260,364,478,604,741,870,975,1071,106,177,247,248,350,466,594,732,861,969,1065,71,111,199,262,366,480,606,743,866,974,1070,143,147,149,246,349,465,593,731,860,968,1064,153,157,200,272,368,482,608,736,865,973,1069,70,72,146,243,344,459,587,727,858,967,1063,85,168,174,271,376,484,597,735,864,972,1068)(68,137,230,330,443,572,711,843,952,1049,1134,233,145,240,341,448,577,716,847,956,1053,1138,251,267,245,347,462,586,725,855,960,1057,1142,138,140,234,332,445,574,713,844,953,1050,1135,151,172,242,342,456,578,717,848,957,1054,1139,194,155,254,348,463,591,726,856,965,1058,1143,139,231,235,333,446,575,714,845,954,1051,1136,238,241,269,343,457,584,718,849,958,1055,1140,252,255,256,359,464,592,730,857,966,1062,1144,142,198,274,334,447,576,715,846,955,1052,1137,167,244,259,363,458,585,724,850,959,1056,1141)(136,226,322,431,554,690,828,940,1037,1122,1195,324,268,374,477,590,729,859,951,1047,1131,1203,354,253,356,442,570,707,839,947,1043,1127,1199,232,328,378,455,583,723,854,964,1061,1145,1206,250,339,362,461,589,728,842,950,1046,1130,1202,292,327,329,441,568,706,838,946,1042,1126,1198,228,293,340,454,582,722,853,963,1060,1133,1205,337,345,346,460,588,710,841,949,1045,1129,1201,227,229,326,438,565,702,835,945,1041,1125,1197,237,239,338,453,581,721,852,962,1048,1132,1204,266,357,358,473,571,709,840,948,1044,1128,1200)(225,318,424,545,678,814,928,1025,1110,1183,1244,427,435,437,563,699,831,939,1035,1119,1191,1251,319,321,429,551,686,823,934,1030,1114,1186,1246,331,373,472,564,700,832,941,1036,1120,1192,1252,353,355,430,552,687,824,935,1031,1115,1187,1247,394,436,490,600,701,833,942,1038,1121,1193,1253,320,425,471,553,688,825,936,1032,1116,1188,1248,452,439,492,603,708,834,943,1039,1123,1194,1254,323,395,476,569,689,826,937,1033,1117,1189,1249,336,451,440,567,705,837,944,1040,1124,1196,1255,372,325,434,561,696,827,938,1034,1118,1190,1250)(317,420,537,666,799,914,1013,1098,1171,1232,1281,539,559,566,685,821,931,1027,1111,1182,1242,1290,426,543,560,695,829,927,1023,1107,1179,1239,1287,444,542,544,676,811,924,1020,1104,1176,1236,1284,470,508,562,698,830,933,1029,1113,1185,1245,1292,507,549,550,684,819,930,1026,1109,1181,1241,1289,422,433,558,693,813,926,1022,1106,1178,1238,1286,421,423,541,673,807,920,1017,1102,1175,1235,1283,432,556,616,704,822,932,1028,1112,1184,1243,1291,450,428,548,682,816,929,1024,1108,1180,1240,1288,488,489,615,677,812,925,1021,1105,1177,1237,1285)(419,533,659,791,904,1001,1086,1159,1220,1269,1306,534,536,663,794,907,1004,1089,1162,1223,1272,1308,546,664,665,797,910,1007,1092,1165,1225,1273,1309,573,671,703,798,912,1010,1094,1166,1226,1274,1310,599,674,694,820,913,1011,1095,1167,1227,1275,1311,631,632,675,809,921,1012,1096,1168,1228,1276,1312,535,660,697,810,922,1018,1097,1169,1229,1277,1313,538,557,683,818,923,1019,1103,1170,1230,1278,1314,555,540,669,802,916,1014,1099,1172,1231,1279,1315,580,670,672,805,918,1015,1100,1173,1233,1280,1316,614,547,681,806,919,1016,1101,1174,1234,1282,1317)(532,656,787,900,996,1080,1152,1212,1260,1296,1320,661,789,790,902,998,1082,1154,1214,1262,1298,1322,679,692,793,905,1000,1084,1156,1216,1264,1300,1324,712,668,796,908,1003,1087,1158,1218,1266,1302,1326,738,803,836,915,1006,1090,1161,1221,1268,1304,1328,765,680,815,917,1009,1093,1164,1224,1271,1307,1330,657,658,788,901,997,1081,1153,1213,1261,1297,1321,667,662,792,903,999,1083,1155,1215,1263,1299,1323,691,795,817,906,1002,1085,1157,1217,1265,1301,1325,720,800,801,909,1005,1088,1160,1219,1267,1303,1327,751,808,804,911,1008,1091,1163,1222,1270,1305,1329)Polycyclic presentation:
11^4|b^11=d^2,b^a=bc,c^a=cd241 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(14641, 10);