Finite Group DatabasebetaFinite Group Databasebeta| order | 24642 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(D1369⋊C3) |
| unambiguous semidirect product | (C1369⋊C3)⋊D3 |
| unambiguous semidirect product | D4107⋊C3 |
| Fitting, derived | C3×C1369 ⋊ C2×C3 |
| Hall | C3×C1369⋊C3 ⋊ C2 |
| Sylow | C1369 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C1369 | C2×C3 |
| Frattini | C37 | C3⋊(D37⋊C3) |
| socle | C3×C37 | D37⋊C3 |
| LCS | G ▷ C3×C1369 | |
| UCS | C1 | |
27 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1372 |
| transitive degree | μt(G) = 4107 |
A minimal faithful permutation representation — 2 generators on 1372 points:
(2,11,13,4,5,14)(3,23,25,8,9,27)(6,41,43,16,17,101)(7,53,54,20,21,131)(10,70,72,28,29,163)(12,62,82,24,33,97)(15,95,64,38,39,153)(18,112,114,46,47,247)(19,124,125,50,51,276)(22,139,140,57,58,309)(26,84,35,65,66,87)(30,174,176,75,76,374)(31,106,89,77,44,36)(32,135,136,80,55,68)(34,166,192,71,85,358)(37,60,134,92,93,56)(40,216,168,98,99,362)(42,151,229,63,104,243)(45,240,154,109,110,340)(48,259,158,117,118,344)(49,269,270,121,122,513)(52,287,288,128,129,545)(59,320,321,143,144,596)(61,169,161,148,74,69)(67,231,107,156,157,235)(73,365,88,170,171,198)(78,253,254,179,115,221)(79,281,282,182,126,294)(81,314,315,184,141,353)(83,393,200,187,188,197)(86,376,409,175,195,666)(90,102,252,201,202,223)(91,132,279,205,206,283)(94,164,312,208,209,319)(96,242,391,178,213,186)(100,442,378,219,220,669)(103,280,347,226,227,159)(105,360,458,167,232,644)(108,211,301,237,238,426)(111,475,363,244,245,649)(113,120,484,152,250,267)(116,490,341,256,257,625)(119,464,189,263,264,395)(123,523,524,273,274,863)(127,222,146,284,285,327)(130,556,557,291,292,912)(133,313,349,298,299,160)(137,190,234,303,304,398)(138,212,364,306,307,430)(142,295,147,317,318,330)(145,605,199,324,325,416)(149,380,381,333,177,354)(150,403,351,335,218,339)(155,367,408,332,194,215)(162,165,379,352,334,357)(172,655,236,368,369,465)(173,191,302,372,373,400)(180,499,305,384,261,481)(181,517,518,386,271,530)(183,550,551,388,289,581)(185,599,600,390,322,663)(193,698,417,404,405,415)(196,664,210,375,412,431)(203,248,345,420,346,249)(204,277,516,423,424,532)(207,310,548,427,428,583)(214,506,507,383,266,456)(217,477,695,382,440,402)(224,630,452,397,449,445)(225,421,496,396,450,563)(228,767,769,453,454,637)(230,505,418,385,265,414)(233,667,783,377,461,1004)(239,438,589,467,468,745)(241,492,682,419,472,392)(246,805,670,479,480,1007)(251,646,815,361,485,850)(255,470,575,487,488,793)(258,825,650,493,494,991)(260,422,514,338,497,660)(262,658,626,501,370,974)(268,847,848,510,511,973)(272,482,328,520,521,610)(275,873,874,527,528,1207)(278,549,632,535,536,348)(286,471,651,542,543,796)(290,531,329,553,554,613)(293,922,466,560,561,790)(296,857,757,566,519,446)(297,906,909,569,552,564)(300,934,935,572,573,638)(308,439,671,579,580,719)(311,598,634,586,587,350)(316,652,463,591,592,768)(323,582,331,602,603,615)(326,653,576,607,608,748)(336,755,673,588,444,648)(337,777,635,620,478,624)(342,714,715,618,411,642)(343,657,782,617,460,474)(355,905,759,639,555,447)(356,717,929,627,604,565)(359,709,636,574,643,623)(366,606,744,616,654,437)(371,457,538,661,662,774)(387,867,868,678,525,897)(389,916,917,680,558,955)(394,700,707,619,685,681)(399,539,908,689,690,571)(401,590,958,693,694,609)(406,1036,504,701,702,789)(407,684,577,705,706,692)(410,1042,722,710,711,721)(413,595,434,672,718,740)(425,546,866,728,729,899)(429,597,914,733,734,956)(432,541,761,735,727,448)(433,594,891,738,732,540)(435,697,578,741,742,731)(436,1014,933,676,725,814)(441,1012,1085,675,750,776)(443,807,1039,674,753,708)(451,1067,1102,764,765,941)(455,1106,1107,771,772,1001)(459,1112,723,778,779,720)(462,1002,760,665,786,1097)(469,752,716,791,792,1051)(473,1000,885,688,799,773)(476,827,1031,724,803,696)(483,736,859,811,812,980)(486,1005,1158,668,818,1291)(489,801,952,820,821,1142)(491,659,855,631,515,683)(495,1169,1008,828,829,1170)(498,988,1172,647,830,1218)(500,823,937,832,593,996)(502,877,878,835,529,1216)(503,756,533,622,836,704)(508,1111,939,841,842,1131)(509,629,611,844,845,817)(512,1188,1189,851,852,1284)(522,824,992,860,861,1165)(526,854,612,870,871,816)(534,1020,834,703,883,927)(537,1225,1226,886,887,981)(544,802,1009,895,896,1128)(547,915,978,902,903,633)(559,898,614,919,920,966)(562,997,889,924,925,1145)(567,971,1028,781,839,809)(568,1115,1129,780,869,880)(570,1244,1245,931,918,942)(584,1203,1095,944,945,758)(585,1242,1251,948,949,928)(601,963,788,959,960,1090)(621,1151,1011,904,808,990)(628,1123,1124,968,785,985)(640,1205,1153,982,872,810)(641,1126,1220,975,921,881)(645,1118,979,888,986,972)(656,923,1141,967,998,800)(677,1114,1038,970,849,1019)(679,1210,1211,1017,875,1239)(686,1264,1050,950,1022,1030)(687,964,1116,969,1023,840)(691,907,1243,1026,1027,951)(699,1043,1073,932,1035,1018)(712,1306,838,1044,1045,1130)(713,1034,938,1048,1049,1081)(726,864,1186,1059,879,1201)(730,913,1209,1063,1064,1240)(737,910,1204,1068,1062,882)(739,962,1247,1070,1066,943)(743,1093,1054,1071,1072,1074)(746,1237,1098,1075,1076,762)(747,1053,1228,1079,1080,892)(749,1041,940,1082,1083,1065)(751,1296,1265,1015,1087,1157)(754,1295,1329,1013,1091,1117)(763,794,893,1100,770,993)(766,1321,1336,1103,1104,1268)(775,1105,1249,1088,1109,926)(784,1341,1056,1119,1120,1055)(787,911,797,1010,1127,1136)(795,954,1196,1133,1134,858)(798,1033,894,1137,1138,1061)(804,1182,1236,1025,1147,1110)(806,1171,1304,1057,1149,1040)(813,1077,1357,1155,1156,1290)(819,1006,1154,1003,831,1292)(822,1148,1052,1162,1163,1150)(826,853,1202,977,865,1032)(833,1167,1266,1174,961,1248)(837,1132,1238,989,1176,1286)(843,1356,1311,1180,1181,1354)(846,999,1217,1183,1184,1289)(856,1302,1198,1193,1194,1280)(862,1168,1293,1199,1060,1208)(876,1200,965,1213,1214,1276)(884,1368,1361,1222,1223,1259)(890,1254,1255,1084,936,994)(900,1308,1333,1047,1233,1094)(901,1301,1175,1046,1234,1250)(930,1364,1365,1252,1212,1231)(946,1344,1323,1122,1261,1152)(947,1343,1348,1121,1262,1219)(953,1273,1269,1029,995,1021)(957,1177,1335,1270,1271,1096)(976,983,1359,1277,1160,1187)(984,1173,1312,1159,1215,1192)(987,1358,1279,1195,1191,1278)(1016,1362,1309,1235,1190,1300)(1024,1281,1345,1263,1298,1324)(1037,1307,1353,1253,1303,1179)(1058,1185,1257,1314,1256,1287)(1069,1246,1366,1318,1317,1232)(1078,1346,1367,1322,1272,1221)(1086,1340,1258,1325,1326,1349)(1089,1125,1331,1313,1092,1099)(1101,1143,1337,1330,1108,1275)(1113,1342,1319,1224,1339,1297)(1135,1274,1369,1350,1351,1260)(1139,1332,1347,1328,1352,1320)(1140,1338,1229,1327,1227,1285)(1144,1310,1363,1334,1241,1197)(1146,1305,1230,1315,1355,1316)(1161,1206,1166,1294,1360,1282)(1164,1267,1283,1299,1288,1178)(1371,1372);(1,175,1009,631,811,1281,1221,420,1063,1243,215,335,310,1311,424,645,405,545,1176,794,920,685,537,541,1142,948,62,744,936,425,270,1341,1295,249,826,787,34,8,375,670,346,1155,1307,882,497,1167,917,97,588,594,1285,703,360,1049,536,1135,773,985,477,278,894,1134,686,66,596,601,204,871,1339,1101,112,1170,462,12,28,672,633,158,1005,1045,532,1059,1185,1018,39,311,589,993,1222,657,943,454,795,868,644,1082,124,1230,944,394,373,587,429,90,1312,1190,763,256,1208,233,4,75,671,348,845,983,1197,450,1168,1157,681,307,139,1054,704,1113,369,583,1109,470,1365,478,890,284,1248,1121,187,693,573,207,414,1291,1171,449,493,1161,105,20,174,378,159,1277,1217,858,1014,825,1336,391,740,317,994,406,778,925,552,797,456,1324,245,539,542,1288,1024,304,664,322,91,720,1191,833,253,828,819,42,57,380,350,67,984,878,519,1314,847,1297,238,590,312,652,1253,567,911,289,471,1116,1085,896,281,895,1122,687,689,604,210,36,1055,852,500,487,873,486,16,143,379,160,617,640,862,271,1146,815,1019,738,314,718,407,930,937,555,1027,240,1353,842,889,279,922,784,395,1026,323,94,197,1089,829,261,820,1213,251,50,324,176,68,968,992,522,1000,803,1359,682,1070,313,653,193,568,1257,290,743,229,1239,821,538,891,998,766,224,1051,144,37,415,1332,502,252,1162,1173,113,128,607,162,26,641,650,272,1327,1114,1200,735,953,140,365,932,296,1320,129,435,782,1236,1334,280,1228,785,451,473,1080,100,14,721,1305,262,761,1210,831,46,291,958,69,332,355,1189,122,1086,779,863,1068,591,603,194,570,575,1076,902,213,1141,1227,1094,125,1042,461,225,804,1066,40,87,752,1040,118,1098,1364,498,121,560,600,27,618,651,1280,659,750,1344,855,1318,908,642,85,297,939,1062,621,188,912,1108,756,554,1035,455,102,1150,734,15,198,1093,1250,114,1331,1343,260,273,924,623,9,356,363,980,1193,964,1232,812,1164,551,358,574,132,1258,729,337,706,903,1096,445,1220,1022,228,241,1241,674,5,416,1041,927,521,1321,1177,117,527,1249,339,148,164,1011,660,987,702,899,830,823,1245,218,300,301,1275,1046,151,1269,887,758,518,1004,807,103,476,1351,382,80,442,708,563,1153,1368,837,250,877,1107,153,333,364,981,370,646,1308,883,1166,814,1030,99,133,578,1286,1016,157,956,772,446,1129,986,788,41,806,1270,178,184,755,928,305,1292,1301,503,510,864,972,93,165,168,637,1184,611,1260,765,824,1102,695,580,53,940,1047,677,662,949,760,221,1348,808,463,109,913,1013,77,390,709,564,1028,1077,1209,263,851,1172,624,330,70,673,371,965,626,945,1296,490,1361,742,576,134,963,712,385,1088,918,447,464,1329,480,234,244,1246,675,182,376,409,294,1323,1302,1178,506,865,1357,340,615,169,638,172,612,1186,869,1139,484,1300,468,302,306,995,679,180,1002,558,222,789,1326,466,106,479,1346,383,388,714,565,137,1078,1032,839,841,523,1278,327,366,72,353,967,328,1206,525,798,844,1031,1079,135,579,713,387,436,921,469,101,1130,1181,236,237,556,1125,179,680,606,295,692,746,874,505,1180,870,973,613,170,381,173,614,341,872,1182,472,1183,1138,938,54,605,410,181,751,559,239,235,1148,1163,107,467,919,1015,386,710,325,131,1081,1033,846,492,1147,1205,625,966,372,354,73,329,848,526,843,265,1207,1075,577,318,654,389,78,1092,292,108,465,1356,838,43,791,1126,676,678,1048,308,55,747,696,509,1137,867,1360,610,656,315,163,616,146,1279,274,508,971,1202,1322,303,929,411,183,214,1272,246,44,790,1340,504,285,916,786,384,1017,1273,138,400,438,1309,267,1328,1115,1201,816,368,935,74,331,154,1290,853,266,1283,1194,1152,282,666,195,79,441,1317,111,398,805,1117,264,759,1244,775,230,1044,960,56,748,697,1259,257,1087,1203,974,1276,661,648,29,147,635,1218,512,119,1240,1156,809,909,643,185,31,754,1064,45,768,1151,1219,254,1097,1242,457,970,900,1065,21,439,402,941,861,1067,1369,817,999,769,362,334,60,979,879,268,622,1234,1154,481,1251,444,81,96,957,1057,17,1090,1118,880,757,1106,914,231,1235,1238,731,298,216,1050,933,1282,1020,1233,485,658,1226,430,149,64,1001,529,120,989,1223,810,496,1039,220,32,217,1274,724,226,753,783,530,1095,1225,1021,104,901,1337,426,572,403,942,593,988,866,1036,1195,514,990,209,61,351,926,275,48,1271,1104,482,834,1083,199,11,443,1310,419,453,1264,881,452,1335,915,684,620,546,1349,206,359,192,581,1299,736,1366,1023,856,859,649,627,23,636,562,123,338,1262,1099,247,1175,1072,88,38,597,1052,202,771,1043,531,836,1143,557,393,904,910,1131,569,166,715,571,1069,683,1261,1012,515,1198,796,342,25,663,293,49,647,1212,762,259,1304,792,35,98,962,1025,396,667,711,276,1333,1338,800,242,547,1237,793,931,367,582,316,737,524,1112,1325,269,1284,639,155,161,609,130,18,1006,875,448,501,1355,722,13,219,1053,688,764,1123,892,227,1144,1110,474,741,287,1347,566,699,171,309,1029,432,1214,849,1140,520,991,975,65,352,326,52,152,1215,822,223,835,1352,417,92,320,716,397,1103,923,540,774,801,1211,243,1071,553,1287,780,404,608,299,739,392,1187,827,799,860,1165,628,136,374,145,19,361,876,489,499,1169,1313,200,208,602,691,189,1119,561,283,1145,1111,1179,110,907,905,996,1252,705,595,141,433,1038,850,1315,517,1199,982,343,349,357,59,6,668,528,255,832,1188,1056,89,431,717,399,969,946,544,126,802,776,840,543,550,1127,781,1037,592,319,694,211,1319,511,1058,857,1216,1159,156,634,177,22,63,1003,495,115,1174,1358,723,205,599,412,190,1263,1267,286,690,475,1345,507,1136,906,997,459,701,1255,142,434,186,1268,494,725,1196,1289,976,347,669,76,7,167,1294,258,630,1149,1158,418,427,934,401,83,947,1266,127,1084,777,1231,488,1105,548,655,1224,533,1074,58,212,707,1265,1060,421,1363,1160,629,632,719,30,2,377,1293,116,1100,1362,1192,201,733,598,191,619,584,1316,51,749,458,897,1133,767,1247,460,884,893,745,586,95,1073,1256,726,516,1306,818,344,978,413,10,24,665,1008,47,1330,1342,854,423,959,321,84,950,954,1061,535,440,1124,885,1350,549,1034,232,534,1229,732,336,82,955,961,422,1204,1303,813,248,1007,196,3,71,1010,977,345,1091,1120,513,728,1254,437,33,585,952,727,886,700,898,770,1132,288,698,888,277,1354,428,150,408,951,730,203,1367,1298,483,491,1128,86)(1370,1371,1372)Polycyclic presentation:
2,3^2,37^2|d^37=e^36,c^a=c^2,d^a=d^36e,e^a=e^36,d^b=d^26e^22,e^b=e^26693 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24642, 12);