Finite Group DatabasebetaFinite Group Databasebeta| order | 20006 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | D1429⋊C7 |
| unambiguous semidirect product | (C1429⋊C7)⋊C2 |
| Fitting, socle, derived, Sylow | C1429 ⋊ C2×C7 |
| Hall | C1429⋊C7 ⋊ C2 |
| Hall | D1429 ⋊ C7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C1429 | C2×C7 |
| LCS | G ▷ C1429 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1429 |
| transitive degree | μt(G) = 1429 |
A minimal faithful permutation representation — 2 generators on 1429 points:
(2,6,7,25,30,116,139,3,4,12,14,54,65,244)(5,19,20,88,92,386,400,9,10,41,43,187,194,714)(8,35,36,159,164,628,289,16,17,75,77,330,340,563)(11,48,49,217,221,55,245,22,23,103,105,444,117,118)(13,59,60,261,266,859,889,27,28,126,128,473,522,490)(15,70,71,312,316,290,291,32,33,150,152,603,140,565)(18,82,83,358,363,1057,738,38,39,174,176,664,670,1062)(21,97,98,419,422,798,564,45,46,204,206,772,779,968)(24,110,111,465,469,347,348,51,52,231,233,846,168,649)(26,120,121,496,498,73,74,56,57,249,251,897,34,157)(29,133,134,537,370,987,1078,62,63,276,278,684,685,1109)(31,144,145,581,66,293,294,67,68,299,301,141,142,570)(37,169,170,651,353,354,566,79,80,349,351,171,637,658)(40,181,182,690,694,991,491,85,86,373,375,1082,1087,891)(42,189,190,724,728,789,1261,89,90,391,393,1116,1119,890)(44,199,200,752,756,567,568,94,95,409,411,1142,292,972)(47,211,212,435,436,837,838,100,101,432,213,801,461,669)(50,226,136,137,550,552,969,107,108,64,283,284,950,921)(53,238,239,868,872,1143,1206,113,114,479,481,1219,1023,821)(58,255,256,907,909,715,996,123,124,504,506,1275,587,1040)(61,270,271,925,928,1031,158,130,131,525,527,1194,976,328)(69,306,307,1004,195,739,740,147,148,594,596,401,402,1127)(72,321,322,590,341,352,571,154,155,616,617,638,639,656)(76,334,335,871,1036,1211,624,161,162,632,633,1248,827,327)(78,345,260,323,607,572,573,166,167,512,513,619,295,655)(81,356,262,382,876,1104,899,172,173,516,517,708,1063,1267)(84,368,369,1075,315,433,971,178,179,680,682,604,605,797)(87,380,381,1098,1101,620,650,184,185,704,706,1264,1024,1025)(91,395,396,965,966,1051,1394,191,192,733,562,654,1171,1354)(93,404,405,895,143,575,576,196,197,495,745,296,297,980)(96,416,417,1157,1161,269,900,202,203,765,767,1370,499,524)(99,427,428,1179,538,539,973,208,209,786,788,277,939,796)(102,438,439,1080,843,844,892,214,215,805,689,464,1225,1226)(104,300,446,186,515,710,1237,218,219,582,383,384,862,1106)(106,452,453,188,718,719,974,223,224,829,388,389,569,1113)(109,459,460,547,548,1170,668,228,229,339,282,773,948,362)(112,474,475,1017,657,357,920,235,236,606,806,1050,662,520)(115,484,485,1254,1214,390,735,241,242,878,880,1396,721,722)(119,492,258,259,716,717,1076,246,247,125,510,387,1110,737)(122,500,493,494,320,1021,817,252,253,248,894,614,615,623)(127,519,359,483,1061,1130,1285,263,264,240,666,743,923,970)(129,523,455,456,1218,818,819,267,268,225,833,834,445,602)(132,531,532,1293,951,952,1039,273,274,933,935,551,647,636)(135,544,545,1190,842,1188,329,280,281,944,802,1222,1223,626)(138,557,558,1256,1312,835,1306,286,287,959,961,1419,953,1048)(146,588,589,342,343,346,584,303,304,643,165,641,642,536)(149,533,600,1296,450,151,205,309,310,936,1014,826,313,314)(153,611,612,608,609,585,586,318,319,613,317,618,302,514)(156,511,621,1114,671,888,954,324,325,720,1026,1064,553,1260)(160,629,578,579,1324,222,521,331,332,298,985,986,265,451)(163,556,635,1309,734,1326,1384,337,338,958,1038,1123,1124,1352)(175,237,667,863,1355,1181,1027,360,361,477,1052,1238,1369,625)(177,675,676,993,994,977,978,365,366,1071,583,1328,574,938)(180,687,688,1245,1286,1220,1221,371,372,867,1079,1416,839,1086)(183,699,700,1277,1128,1330,918,377,378,995,1094,997,1305,518)(193,509,665,1279,1360,1320,457,397,398,913,1059,1299,1189,227)(198,749,750,1152,403,1129,1053,407,408,1035,762,741,742,1357)(201,760,761,1374,640,1195,981,413,414,1115,1151,1041,807,1047)(207,783,243,883,884,982,983,424,425,487,488,686,577,1045)(210,793,794,849,850,713,1268,430,431,1184,468,784,1107,1108)(216,811,812,1013,1244,1259,1049,441,442,1199,1193,1340,1415,1081)(220,595,822,1274,1117,1420,1186,447,448,1005,1208,1346,1288,1399)(230,840,841,627,554,555,1262,462,463,778,1029,285,955,956)(232,847,848,1333,877,1103,1387,466,467,1230,1231,707,1138,1380)(234,856,857,250,785,898,1321,471,472,1235,426,497,975,1266)(254,902,903,1034,759,1359,736,501,502,631,1156,1058,1149,399)(257,911,912,1088,916,421,622,507,508,723,1278,1169,777,326)(272,930,489,852,659,660,601,528,529,886,887,355,1022,311)(275,308,729,1010,780,781,893,534,535,598,851,423,1174,1175)(279,941,701,1258,1257,1205,1173,541,542,881,1095,1389,820,776)(288,333,964,1033,1006,1421,1111,560,561,630,1164,1335,1336,1207)(305,998,999,672,673,1102,1134,591,592,1003,364,1066,1067,1379)(336,963,1037,1406,1234,1056,922,559,634,1284,1224,1401,663,1030)(344,795,1043,1311,406,1018,1135,644,645,1139,1007,746,747,1345)(350,945,1044,790,791,1341,1212,652,653,646,429,1182,1183,828)(367,1000,580,683,1327,853,854,678,679,988,989,1077,470,885)(374,709,1084,1381,1424,860,861,691,692,1105,1372,1400,476,1028)(376,1091,873,1334,1112,1232,990,696,697,1249,1250,1383,979,1300)(379,1096,992,814,815,1287,1085,702,703,727,443,926,1202,693)(385,924,836,823,824,1297,1126,711,712,458,449,937,1083,1385)(392,845,1118,1409,914,1072,1351,725,726,1228,1269,1280,1281,1319)(394,597,1121,412,1147,1148,1019,730,731,1008,757,758,830,610)(410,1144,1145,1251,1342,1343,1215,753,754,1020,1229,1015,1427,831)(415,1154,1155,1404,1065,1180,1307,763,764,1361,1217,787,957,1397)(418,1165,1166,1068,929,1313,1046,769,770,1348,1363,962,1290,648)(420,546,1172,1197,1423,454,940,774,775,946,1395,1325,540,832)(434,808,809,1376,1329,1322,1323,799,440,1196,1097,1125,984,1302)(437,1192,864,1358,599,1338,1339,803,804,1239,1240,1011,1012,1378)(478,1242,1243,1255,1282,1331,1422,865,866,879,1133,1418,1386,1298)(480,1246,1247,1158,1159,503,1271,869,870,1216,766,901,904,905)(482,1252,1210,931,932,543,1301,874,875,825,530,1292,942,943)(486,698,1177,1131,1146,1016,919,681,882,1093,755,1060,1344,855)(505,1150,1276,1203,1120,1308,1411,792,908,1347,816,1291,1318,1236)(526,813,1289,1402,1350,858,1054,800,927,1201,1413,1425,1213,661)(549,1168,967,1055,934,1227,1263,771,949,1316,1317,1191,1294,1407)(593,677,1332,1089,1090,1042,1032,1001,1002,1073,695,1373,1353,1200)(674,1365,1366,1362,748,1349,705,1069,1070,1428,1153,1136,1137,1099)(732,1209,896,1337,1162,1163,1375,910,1122,1009,1265,768,1390,1391)(744,1272,1273,917,1408,1414,1187,1132,906,1371,1283,1241,1412,915)(751,1356,1167,1176,1160,1270,1310,1140,1141,1392,782,1388,1295,960)(810,1368,1382,1405,1185,1314,1377,1074,1198,1417,1429,1364,1398,1100)(947,1367,1403,1393,1092,1204,1410,1303,1304,1315,1426,1178,1253,1233);(1,2,5,11,24,53,115,243,489,888,550,77,165,344,646,1045,740,954,1223,779,605,101,213,437,592,1000,1331,742,472,858,1212,521,922,1029,626,328,157,327,625,1028,1349,1273,1245,1338,1220,837,459,492,893,1263,1397,1357,573,651,1050,639,349,134,279,543,898,719,567,290,116,73,32,12,26,58,125,260,514,916,422,527,929,959,632,794,1186,1394,1078,400,738,838,668,360,39,84,180,297,577,984,1151,1373,1429,1404,940,660,1021,1056,662,356,661,1055,1358,1379,895,844,1227,1408,1239,667,902,1270,1293,1179,1285,291,566,971,918,516,60,129,269,496,386,715,585,299,202,94,41,71,153,320,164,341,640,946,1201,1403,630,958,428,789,1181,682,1076,736,397,191,89,10,21,47,102,216,443,816,1204,1252,1359,1306,348,650,1049,1080,687,854,883,1259,1290,1345,442,813,1069,672,358,665,1060,1014,1341,1344,819,601,309,147,67,25,55,119,248,495,896,731,594,455,223,103,30,66,143,298,580,990,542,809,1197,1051,548,206,423,782,1177,1396,1148,224,454,538,63,135,282,549,769,416,766,1160,1103,1136,746,404,744,1133,1376,1096,900,1268,1339,1377,1099,185,382,709,989,680,396,735,1126,1276,1145,1019,608,312,347,166,75,160,333,631,1035,1121,1342,936,266,471,231,249,417,768,1164,676,974,1320,436,802,1191,274,533,937,1163,880,1257,756,994,1329,1123,192,325,510,915,645,793,1185,1363,1066,1362,1152,760,1150,1246,1417,1084,1292,1410,1010,849,466,52,112,237,478,867,190,394,732,907,752,1143,609,616,678,365,174,359,238,480,871,728,1120,529,708,874,479,504,200,412,759,363,671,1065,209,429,792,124,257,509,914,1282,1094,1275,1336,1095,1335,448,148,308,599,1013,1192,1400,1266,1332,1426,1392,1165,785,847,1089,690,1083,1250,596,1007,1310,287,559,334,407,196,88,188,390,723,1115,1383,389,720,882,696,373,261,515,917,1284,1416,712,268,378,701,878,716,92,195,403,743,1131,753,95,201,415,338,554,283,551,953,833,1219,1391,697,1092,850,783,1168,1238,1413,1368,1301,1231,999,1001,591,303,144,300,583,995,822,1209,1208,1272,904,501,252,120,250,424,204,420,776,1059,1361,1427,754,508,483,877,704,525,926,1288,961,1243,242,486,179,370,686,619,603,834,758,481,873,1251,726,843,462,228,107,48,104,220,449,825,1161,1327,1425,1248,1182,764,1156,1366,1070,1367,1353,589,641,1024,513,321,189,392,727,446,821,1207,1110,1157,931,528,270,526,799,432,796,972,1319,1309,1409,1159,253,236,476,862,933,1118,1337,1141,863,966,652,80,171,355,319,613,453,831,1216,694,1088,1124,685,985,702,377,181,374,693,1086,1267,897,986,1325,1419,1043,408,751,1067,1098,1174,1087,301,584,534,273,130,59,127,265,217,445,820,1026,1348,1303,775,544,945,1195,805,537,938,1298,956,1226,891,118,108,227,458,598,724,1117,747,475,604,313,33,72,156,326,623,221,450,827,522,923,1281,366,677,1074,808,438,806,1194,1401,887,846,576,981,1046,642,345,647,670,1063,1107,711,383,184,85,38,16,6,13,29,64,138,288,562,967,1318,1180,955,629,1032,1199,944,1302,1330,552,841,777,314,46,99,210,304,590,997,402,233,470,855,264,520,921,968,563,244,490,890,1106,1380,905,866,1244,1406,1222,840,894,1264,1414,1411,1235,294,571,976,340,637,939,1299,1372,848,1232,977,572,293,172,79,35,76,163,339,636,1040,1352,948,1202,1398,634,963,1314,1287,1170,1384,996,1039,460,337,161,17,37,81,142,295,574,979,1230,1381,1360,539,354,628,1031,656,570,857,1236,1412,1101,494,463,842,1224,1340,1242,1271,1387,1237,1261,889,139,289,564,969,920,519,919,853,465,739,1128,617,588,430,208,97,205,421,778,950,1305,1323,545,812,1200,332,555,957,1308,1316,965,560,286,136,62,27,4,8,18,40,87,186,385,713,1104,1057,952,167,346,648,1047,980,469,852,1234,928,1017,215,440,810,1002,675,1072,1130,859,1211,826,444,817,622,324,154,70,151,315,606,1018,1346,393,729,836,457,226,245,491,892,1262,1422,978,278,439,807,653,281,546,947,1166,1114,1205,818,105,222,263,28,61,132,275,536,581,991,781,706,1102,1140,749,1139,1312,1423,1324,498,899,1221,1085,691,86,183,379,579,987,1326,1278,1082,1247,1215,829,612,611,659,353,169,350,654,1052,1356,1265,1228,532,710,860,474,500,901,1269,1038,1351,568,973,212,434,800,131,272,530,767,717,1111,1206,582,992,725,90,155,323,620,343,643,1042,1304,1365,1428,1034,1154,791,1036,1350,1077,1370,1210,823,447,218,23,50,109,230,464,845,1229,1249,868,1147,1218,316,607,884,684,368,681,484,879,1256,1420,815,271,381,707,666,911,1144,1343,1155,1279,1173,774,98,207,426,57,122,254,503,906,1274,1122,1005,700,993,219,68,146,305,593,1003,1333,943,1198,1402,1355,949,425,784,1178,1091,1375,872,456,835,951,137,285,556,763,413,199,410,755,1061,741,401,187,387,485,881,699,523,924,1286,1037,1283,384,128,182,376,698,621,718,1112,761,912,721,388,43,93,198,162,336,557,960,1311,1004,1334,1297,375,695,467,497,770,1167,1393,1073,1321,1424,804,1193,1011,535,306,595,1006,1258,1421,909,1277,1418,1280,398,507,255,505,790,427,787,1064,664,1058,757,409,256,239,482,876,930,1291,1116,633,869,114,240,83,177,367,322,618,1023,411,506,910,730,391,688,865,361,235,110,232,468,851,1233,932,1105,1382,870,908,414,762,1153,673,1068,1364,431,795,1187,259,511,395,734,1125,1328,1142,1389,1254,1390,824,310,531,934,1190,801,1189,1113,1071,1033,1162,765,121,111,234,473,600,1015,1008,750,903,561,331,36,78,168,152,317,610,1020,1347,1385,722,733,369,683,692,517,380,705,1100,1378,1108,524,703,1097,1255,1132,197,406,748,1138,1388,1158,203,418,771,547,280,133,277,540,452,830,1214,1093,1176,780,419,773,1171,1395,1196,941,1300,988,578,296,141,54,49,106,225,307,597,1009,405,493,246,117,14,31,69,149,311,602,1016,1183,1296,1146,913,176,364,674,927,811,1135,1313,1415,488,885,372,689,1081,1025,649,1048,1149,875,1253,1203,814,441,214,100,45,19,42,91,193,399,737,1075,1369,1119,786,635,964,1315,1289,1172,1041,638,330,614,318,150,20,44,96,145,302,587,194,251,499,267,126,262,518,797,658,565,970,788,935,1295,502,477,864,1241,1294,1225,745,1134,1240,1317,1054,173,357,663,615,1022,832,1217,1405,1090,1374,1322,982,575,371,178,82,175,362,669,1062,714,1109,1354,1399,1184,335,558,962,925,772,1169,586,512,258,123,56,7,15,34,65,140,292,569,975,942,541,276,170,352,657,351,655,1053,1307,1407,1175,247,229,461,839,1012,1079,1371,1137,861,1027,624,74,158,329,627,1030,451,828,1213,856,1129,1386,679,998,803,435,211,433,798,1188,1260,1127,983,1044,644,342,159,284,553,886,487,241,113,51,22,9,3)Polycyclic presentation:
2,7,1429|c^a=c^1428,c^b=c^792116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20006, 1);