Finite Group DatabasebetaFinite Group Databasebeta| order | 14641 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C11×C121 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C1331⋊C11 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C121 | C112 |
| commutator, socle | C11 | C11×C121 |
| prime radical | C112 | C121 |
| LCS | G ▷ C11 ▷ C1 | |
| UCS | C1 ◁ C121 ◁ G | |
| LECS | G ▷ C121 ▷ C11 ▷ C1 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1331 |
| transitive degree | μt(G) = 1331 |
A minimal faithful permutation representation — 2 generators on 1331 points:
(1,2,5,12,26,51,92,151,223,308,406,3,7,14,28,53,94,153,225,310,408,518,9,19,33,58,99,157,229,314,412,521,638,22,42,68,108,165,235,320,418,526,641,756,46,81,122,177,245,327,425,533,646,759,870,86,138,192,259,339,433,541,653,764,873,977,144,209,275,354,446,550,661,771,878,980,1073,215,293,371,462,564,670,779,885,985,1076,1153,299,390,480,581,684,788,893,992,1081,1156,1215,396,500,600,701,802,902,1000,1088,1161,1218,1264,506,621,720,819,916,1009,1096,1168,1223,1267,1300,4,8,15,29,54,95,154,226,311,409,517,10,20,34,59,100,158,230,315,413,522,637,23,43,69,109,166,236,321,419,527,642,755,47,82,123,178,246,328,426,534,647,760,869,87,139,193,260,340,434,542,654,765,874,976,145,210,276,355,447,551,662,772,879,981,1072,216,294,372,463,565,671,780,886,986,1077,1152,300,391,481,582,685,789,894,993,1082,1157,1214,397,501,601,702,803,903,1001,1089,1162,1219,1263,507,622,721,820,917,1010,1097,1169,1224,1268,1299,627,741,838,933,1023,1105,1176,1230,1272,1303,1322,11,21,35,60,101,159,231,316,414,407,519,24,44,70,110,167,237,322,420,528,520,639,48,83,124,179,247,329,427,535,648,640,757,88,140,194,261,341,435,543,655,766,758,871,146,211,277,356,448,552,663,773,880,872,978,217,295,373,464,566,672,781,887,987,979,1074,301,392,482,583,686,790,895,994,1083,1075,1154,398,502,602,703,804,904,1002,1090,1163,1155,1216,508,623,722,821,918,1011,1098,1170,1225,1217,1265,628,742,839,934,1024,1106,1177,1231,1273,1266,1301,746,857,950,1038,1117,1183,1236,1277,1306,1302,1323,25,45,71,111,97,162,241,334,309,410,523,49,84,125,180,163,242,335,441,411,524,643,89,141,195,262,243,336,442,559,525,644,761,147,212,278,357,337,443,560,679,645,762,875,218,296,374,465,444,561,680,797,763,876,982,302,393,483,584,562,681,798,911,877,983,1078,399,503,603,704,682,799,912,1018,984,1079,1158,509,624,723,822,800,913,1019,1113,1080,1159,1220,629,743,840,935,914,1020,1114,1189,1160,1221,1269,747,858,951,1039,1021,1115,1190,1245,1222,1270,1304,861,966,1052,1129,1116,1191,1246,1287,1271,1305,1324,50,85,126,117,103,170,251,224,312,415,529,90,142,196,187,171,252,346,313,416,530,649,148,213,279,270,253,347,454,417,531,650,767,219,297,375,366,348,455,573,532,651,768,881,303,394,484,475,456,574,693,652,769,882,988,400,504,604,595,575,694,811,770,883,989,1084,510,625,724,715,695,812,925,884,990,1085,1164,630,744,841,833,813,926,1031,991,1086,1165,1226,748,859,952,946,927,1032,1124,1087,1166,1227,1274,862,967,1053,1049,1033,1125,1197,1167,1228,1275,1307,969,1064,1138,1137,1126,1198,1251,1229,1276,1308,1325,91,143,65,56,105,173,152,227,317,421,536,149,214,119,106,174,255,228,318,422,537,656,220,298,189,175,256,350,319,423,538,657,774,304,395,272,257,351,458,424,539,658,775,888,401,505,368,352,459,577,540,659,776,889,995,511,626,477,460,578,697,660,777,890,996,1091,631,745,597,579,698,815,778,891,997,1092,1171,749,860,717,699,816,929,892,998,1093,1172,1232,863,968,835,817,930,1035,999,1094,1173,1233,1278,970,1065,948,931,1036,1127,1095,1174,1234,1279,1309,1066,1146,1051,1037,1128,1199,1175,1235,1280,1310,1326,150,133,74,62,114,93,155,232,323,428,544,221,204,130,115,184,156,233,324,429,545,664,305,288,201,185,267,234,325,430,546,665,782,402,385,285,268,363,326,431,547,666,783,896,512,495,382,364,472,432,548,667,784,897,1003,632,616,492,473,592,549,668,785,898,1004,1099,750,736,613,593,712,669,786,899,1005,1100,1178,864,853,733,713,830,787,900,1006,1101,1179,1237,971,963,850,831,943,901,1007,1102,1180,1238,1281,1067,1062,961,944,1047,1008,1103,1181,1239,1282,1311,1147,1145,1061,1048,1136,1104,1182,1240,1283,1312,1327,222,77,31,64,52,96,160,238,330,436,553,306,134,66,118,98,161,239,331,437,554,673,403,205,120,188,164,240,332,438,555,674,791,513,289,190,271,244,333,439,556,675,792,905,633,386,273,367,338,440,557,676,793,906,1012,751,496,369,476,445,558,677,794,907,1013,1107,865,617,478,596,563,678,795,908,1014,1108,1184,972,737,598,716,683,796,909,1015,1109,1185,1241,1068,854,718,834,801,910,1016,1110,1186,1242,1284,1148,964,836,947,915,1017,1111,1187,1243,1285,1313,1210,1063,949,1050,1022,1112,1188,1244,1286,1314,1328,307,39,37,27,55,102,168,248,342,449,567,404,78,75,57,104,169,249,343,450,568,687,514,135,131,107,172,250,344,451,569,688,805,634,206,202,176,254,345,452,570,689,806,919,752,290,286,258,349,453,571,690,807,920,1025,866,387,383,353,457,572,691,808,921,1026,1118,973,497,493,461,576,692,809,922,1027,1119,1192,1069,618,614,580,696,810,923,1028,1120,1193,1247,1149,738,734,700,814,924,1029,1121,1194,1248,1288,1211,855,851,818,928,1030,1122,1195,1249,1289,1315,1260,965,962,932,1034,1123,1196,1250,1290,1316,1329,405,17,13,30,61,112,181,263,358,466,585,515,40,32,63,113,182,264,359,467,586,705,635,79,67,116,183,265,360,468,587,706,823,753,136,121,186,266,361,469,588,707,824,936,867,207,191,269,362,470,589,708,825,937,1040,974,291,274,365,471,590,709,826,938,1041,1130,1070,388,370,474,591,710,827,939,1042,1131,1200,1150,498,479,594,711,828,940,1043,1132,1201,1252,1212,619,599,714,829,941,1044,1133,1202,1253,1291,1261,739,719,832,942,1045,1134,1203,1254,1292,1317,1297,856,837,945,1046,1135,1204,1255,1293,1318,1330,516,6,16,36,72,127,197,280,376,485,605,636,18,38,73,128,198,281,377,486,606,725,754,41,76,129,199,282,378,487,607,726,842,868,80,132,200,283,379,488,608,727,843,953,975,137,203,284,380,489,609,728,844,954,1054,1071,208,287,381,490,610,729,845,955,1055,1139,1151,292,384,491,611,730,846,956,1056,1140,1205,1213,389,494,612,731,847,957,1057,1141,1206,1256,1262,499,615,732,848,958,1058,1142,1207,1257,1294,1298,620,735,849,959,1059,1143,1208,1258,1295,1319,1321,740,852,960,1060,1144,1209,1259,1296,1320,1331);(2,6,17,39,77,133,143,85,45,21,8)(5,13,31,65,71,15,16,37,74,126,35)(7,18,40,78,134,204,214,142,84,44,20)(12,27,56,60,36,64,117,29,30,62,111)(14,32,66,119,125,34,38,75,130,196,70)(19,41,79,135,205,288,298,213,141,83,43)(26,52,97,72,114,101,61,105,54,55,103)(28,57,106,110,73,118,187,59,63,115,180)(33,67,120,189,195,69,76,131,201,279,124)(42,80,136,206,289,385,395,297,212,140,82)(51,93,95,96,159,102,162,112,170,127,173)(53,98,163,128,184,167,113,174,100,104,171)(58,107,175,179,129,188,270,109,116,185,262)(68,121,190,272,278,123,132,202,285,375,194)(81,137,207,290,386,495,505,394,296,211,139)(92,152,197,251,181,241,168,231,160,154,155)(94,156,158,161,237,169,242,182,252,198,255)(99,164,243,199,267,247,183,256,166,172,253)(108,176,257,261,200,271,366,178,186,268,357)(122,191,273,368,374,193,203,286,382,484,277)(138,208,291,387,496,616,626,504,393,295,210)(151,224,248,226,227,263,316,232,280,334,238)(153,228,281,346,264,335,249,322,239,230,233)(157,234,236,240,329,250,336,265,347,282,350)(165,244,337,283,363,341,266,351,246,254,348)(177,258,352,356,284,367,475,260,269,364,465)(192,274,369,477,483,276,287,383,492,604,373)(209,292,388,497,617,736,745,625,503,392,294)(223,309,323,358,311,312,330,376,414,317,342)(225,313,343,315,318,359,420,324,377,441,331)(229,319,378,454,360,442,344,427,332,321,325)(235,326,328,333,435,345,443,361,455,379,458)(245,338,444,380,472,448,362,459,340,349,456)(259,353,460,464,381,476,595,355,365,473,584)(275,370,478,597,603,372,384,493,613,724,482)(293,389,498,618,737,853,860,744,624,502,391)(308,407,415,428,449,485,409,410,421,436,466)(310,411,429,467,413,416,437,486,528,422,450)(314,417,451,419,423,468,535,430,487,559,438)(320,424,488,573,469,560,452,543,439,426,431)(327,432,434,440,552,453,561,470,574,489,577)(339,445,562,490,592,566,471,578,447,457,575)(354,461,579,583,491,596,715,463,474,593,704)(371,479,598,717,723,481,494,614,733,841,602)(390,499,619,738,854,963,968,859,743,623,501)(406,517,519,523,529,536,544,553,567,585,605)(408,520,530,545,568,606,522,524,537,554,586)(412,525,546,587,527,531,555,607,648,538,569)(418,532,570,534,539,588,655,547,608,679,556)(425,540,609,693,589,680,571,663,557,542,548)(433,549,551,558,672,572,681,590,694,610,697)(446,563,682,611,712,686,591,698,565,576,695)(462,580,699,703,612,716,833,582,594,713,822)(480,599,718,835,840,601,615,734,850,952,722)(500,620,739,855,964,1062,1065,967,858,742,622)(518,637,639,643,649,656,664,673,687,705,725)(521,640,650,665,688,726,642,644,657,674,706)(526,645,666,707,647,651,675,727,766,658,689)(533,652,690,654,659,708,773,667,728,797,676)(541,660,729,811,709,798,691,781,677,662,668)(550,669,671,678,790,692,799,710,812,730,815)(564,683,800,731,830,804,711,816,685,696,813)(581,700,817,821,732,834,946,702,714,831,935)(600,719,836,948,951,721,735,851,961,1053,839)(621,740,856,965,1063,1145,1146,1064,966,857,741)(638,755,757,761,767,774,782,791,805,823,842)(641,758,768,783,806,843,760,762,775,792,824)(646,763,784,825,765,769,793,844,880,776,807)(653,770,808,772,777,826,887,785,845,911,794)(661,778,846,925,827,912,809,895,795,780,786)(670,787,789,796,904,810,913,828,926,847,929)(684,801,914,848,943,918,829,930,803,814,927)(701,818,931,934,849,947,1049,820,832,944,1039)(720,837,949,1051,1052,838,852,962,1061,1138,950)(756,869,871,875,881,888,896,905,919,936,953)(759,872,882,897,920,954,874,876,889,906,937)(764,877,898,938,879,883,907,955,987,890,921)(771,884,922,886,891,939,994,899,956,1018,908)(779,892,957,1031,940,1019,923,1002,909,894,900)(788,901,903,910,1011,924,1020,941,1032,958,1035)(802,915,1021,959,1047,1024,942,1036,917,928,1033)(819,932,1037,1038,960,1050,1137,933,945,1048,1129)(870,976,978,982,988,995,1003,1012,1025,1040,1054)(873,979,989,1004,1026,1055,981,983,996,1013,1041)(878,984,1005,1042,986,990,1014,1056,1083,997,1027)(885,991,1028,993,998,1043,1090,1006,1057,1113,1015)(893,999,1058,1124,1044,1114,1029,1098,1016,1001,1007)(902,1008,1010,1017,1106,1030,1115,1045,1125,1059,1127)(916,1022,1116,1060,1136,1117,1046,1128,1023,1034,1126)(977,1072,1074,1078,1084,1091,1099,1107,1118,1130,1139)(980,1075,1085,1100,1119,1140,1077,1079,1092,1108,1131)(985,1080,1101,1132,1082,1086,1109,1141,1163,1093,1120)(992,1087,1121,1089,1094,1133,1170,1102,1142,1189,1110)(1000,1095,1143,1197,1134,1190,1122,1177,1111,1097,1103)(1009,1104,1105,1112,1183,1123,1191,1135,1198,1144,1199)(1073,1152,1154,1158,1164,1171,1178,1184,1192,1200,1205)(1076,1155,1165,1179,1193,1206,1157,1159,1172,1185,1201)(1081,1160,1180,1202,1162,1166,1186,1207,1225,1173,1194)(1088,1167,1195,1169,1174,1203,1231,1181,1208,1245,1187)(1096,1175,1209,1251,1204,1246,1196,1236,1188,1176,1182)(1153,1214,1216,1220,1226,1232,1237,1241,1247,1252,1256)(1156,1217,1227,1238,1248,1257,1219,1221,1233,1242,1253)(1161,1222,1239,1254,1224,1228,1243,1258,1273,1234,1249)(1168,1229,1250,1230,1235,1255,1277,1240,1259,1287,1244)(1215,1263,1265,1269,1274,1278,1281,1284,1288,1291,1294)(1218,1266,1275,1282,1289,1295,1268,1270,1279,1285,1292)(1223,1271,1283,1293,1272,1276,1286,1296,1306,1280,1290)(1264,1299,1301,1304,1307,1309,1311,1313,1315,1317,1319)(1267,1302,1308,1312,1316,1320,1303,1305,1310,1314,1318)(1300,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331)Polycyclic presentation:
11^4|a^11=c,c^11=d,b^a=bd1441 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(14641, 6);