D₃×C₁₄₅₃⋊C₃ #39908526

order	26154
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₁₄₅₃⋊C₃
unambiguous semidirect product	C₃⋊(C₂×C₁₄₅₃⋊C₃)
unambiguous semidirect product	(D₃×C₁₄₅₃)⋊C₃

Splittings

Fitting, socle, derived	C₃×C₁₄₅₃ ⋊ C₂×C₃
Hall	C₃×C₁₄₅₃⋊C₃ ⋊ C₂
Sylow	C₁₄₅₃ ⋊ C₃×D₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₄₅₃	C₂×C₃
LCS	G ▷ C₃×C₁₄₅₃
UCS	C₁

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1456
transitive degree	μ_t(G) = 4359

A minimal faithful permutation representation — 2 generators on 1456 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,70)(41,56,67)(44,59,64)(47,62,61)(50,65,58)(52,68,55)(54,72,92)(57,75,89)(60,78,86)(63,81,83)(66,84,80)(69,87,77)(71,90,74)(73,94,117)(76,97,114)(79,100,111)(82,103,108)(85,106,105)(88,109,102)(91,112,99)(93,115,96)(95,119,145)(98,122,142)(101,125,139)(104,128,136)(107,131,133)(110,134,130)(113,137,127)(116,140,124)(118,143,121)(120,147,176)(123,150,173)(126,153,170)(129,156,167)(132,159,164)(135,162,161)(138,165,158)(141,168,155)(144,171,152)(146,174,149)(148,178,210)(151,181,207)(154,184,204)(157,187,201)(160,190,198)(163,193,195)(166,196,192)(169,199,189)(172,202,186)(175,205,183)(177,208,180)(179,212,247)(182,215,244)(185,218,241)(188,221,238)(191,224,235)(194,227,232)(197,230,229)(200,233,226)(203,236,223)(206,239,220)(209,242,217)(211,245,214)(213,249,287)(216,252,284)(219,255,281)(222,258,278)(225,261,275)(228,264,272)(231,267,269)(234,270,266)(237,273,263)(240,276,260)(243,279,257)(246,282,254)(248,285,251)(250,289,330)(253,292,327)(256,295,324)(259,298,321)(262,301,318)(265,304,315)(268,307,312)(271,310,309)(274,313,306)(277,316,303)(280,319,300)(283,322,297)(286,325,294)(288,328,291)(290,332,376)(293,335,373)(296,338,370)(299,341,367)(302,344,364)(305,347,361)(308,350,358)(311,353,355)(314,356,352)(317,359,349)(320,362,346)(323,365,343)(326,368,340)(329,371,337)(331,374,334)(333,378,425)(336,381,422)(339,384,419)(342,387,416)(345,390,413)(348,393,410)(351,396,407)(354,399,404)(357,402,401)(360,405,398)(363,408,395)(366,411,392)(369,414,389)(372,417,386)(375,420,383)(377,423,380)(379,427,477)(382,430,474)(385,433,471)(388,436,468)(391,439,465)(394,442,462)(397,445,459)(400,448,456)(403,451,453)(406,454,450)(409,457,447)(412,460,444)(415,463,441)(418,466,438)(421,469,435)(424,472,432)(426,475,429)(428,479,532)(431,482,529)(434,485,526)(437,488,523)(440,491,520)(443,494,517)(446,497,514)(449,500,511)(452,503,508)(455,506,505)(458,509,502)(461,512,499)(464,515,496)(467,518,493)(470,521,490)(473,524,487)(476,527,484)(478,530,481)(480,534,590)(483,537,587)(486,540,584)(489,543,581)(492,546,578)(495,549,575)(498,552,572)(501,555,569)(504,558,566)(507,561,563)(510,564,560)(513,567,557)(516,570,554)(519,573,551)(522,576,548)(525,579,545)(528,582,542)(531,585,539)(533,588,536)(535,592,651)(538,595,648)(541,598,645)(544,601,642)(547,604,639)(550,607,636)(553,610,633)(556,613,630)(559,616,627)(562,619,624)(565,622,621)(568,625,618)(571,628,615)(574,631,612)(577,634,609)(580,637,606)(583,640,603)(586,643,600)(589,646,597)(591,649,594)(593,653,715)(596,656,712)(599,659,709)(602,662,706)(605,665,703)(608,668,700)(611,671,697)(614,674,694)(617,677,691)(620,680,688)(623,683,685)(626,686,682)(629,689,679)(632,692,676)(635,695,673)(638,698,670)(641,701,667)(644,704,664)(647,707,661)(650,710,658)(652,713,655)(654,717,781)(657,719,778)(660,722,775)(663,725,772)(666,728,769)(669,731,766)(672,734,763)(675,737,760)(678,740,757)(681,743,754)(684,746,751)(687,749,748)(690,752,745)(693,755,742)(696,758,739)(699,761,736)(702,764,733)(705,767,730)(708,770,727)(711,773,724)(714,776,721)(716,779,718)(720,782,843)(723,784,840)(726,787,837)(729,790,834)(732,793,831)(735,796,828)(738,799,825)(741,802,822)(744,805,819)(747,808,816)(750,811,813)(753,814,810)(756,817,807)(759,820,804)(762,823,801)(765,826,798)(768,829,795)(771,832,792)(774,835,789)(777,838,786)(780,841,783)(785,844,902)(788,846,899)(791,849,896)(794,852,893)(797,855,890)(800,858,887)(803,861,884)(806,864,881)(809,867,878)(812,870,875)(815,873,872)(818,876,869)(821,879,866)(824,882,863)(827,885,860)(830,888,857)(833,891,854)(836,894,851)(839,897,848)(842,900,845)(847,903,958)(850,905,955)(853,908,952)(856,911,949)(859,914,946)(862,917,943)(865,920,940)(868,923,937)(871,926,934)(874,929,931)(877,932,928)(880,935,925)(883,938,922)(886,941,919)(889,944,916)(892,947,913)(895,950,910)(898,953,907)(901,956,904)(906,959,1011)(909,961,1008)(912,964,1005)(915,967,1002)(918,970,999)(921,973,996)(924,976,993)(927,979,990)(930,982,987)(933,985,984)(936,988,981)(939,991,978)(942,994,975)(945,997,972)(948,1000,969)(951,1003,966)(954,1006,963)(957,1009,960)(962,1012,1061)(965,1014,1058)(968,1017,1055)(971,1020,1052)(974,1023,1049)(977,1026,1046)(980,1029,1043)(983,1032,1040)(986,1035,1037)(989,1038,1034)(992,1041,1031)(995,1044,1028)(998,1047,1025)(1001,1050,1022)(1004,1053,1019)(1007,1056,1016)(1010,1059,1013)(1015,1062,1108)(1018,1064,1105)(1021,1067,1102)(1024,1070,1099)(1027,1073,1096)(1030,1076,1093)(1033,1079,1090)(1036,1082,1087)(1039,1085,1084)(1042,1088,1081)(1045,1091,1078)(1048,1094,1075)(1051,1097,1072)(1054,1100,1069)(1057,1103,1066)(1060,1106,1063)(1065,1109,1152)(1068,1111,1149)(1071,1114,1146)(1074,1117,1143)(1077,1120,1140)(1080,1123,1137)(1083,1126,1134)(1086,1129,1131)(1089,1132,1128)(1092,1135,1125)(1095,1138,1122)(1098,1141,1119)(1101,1144,1116)(1104,1147,1113)(1107,1150,1110)(1112,1153,1193)(1115,1155,1190)(1118,1158,1187)(1121,1161,1184)(1124,1164,1181)(1127,1167,1178)(1130,1170,1175)(1133,1173,1172)(1136,1176,1169)(1139,1179,1166)(1142,1182,1163)(1145,1185,1160)(1148,1188,1157)(1151,1191,1154)(1156,1194,1231)(1159,1196,1228)(1162,1199,1225)(1165,1202,1222)(1168,1205,1219)(1171,1208,1216)(1174,1211,1213)(1177,1214,1210)(1180,1217,1207)(1183,1220,1204)(1186,1223,1201)(1189,1226,1198)(1192,1229,1195)(1197,1232,1266)(1200,1234,1263)(1203,1237,1260)(1206,1240,1257)(1209,1243,1254)(1212,1246,1251)(1215,1249,1248)(1218,1252,1245)(1221,1255,1242)(1224,1258,1239)(1227,1261,1236)(1230,1264,1233)(1235,1267,1298)(1238,1269,1295)(1241,1272,1292)(1244,1275,1289)(1247,1278,1286)(1250,1281,1283)(1253,1284,1280)(1256,1287,1277)(1259,1290,1274)(1262,1293,1271)(1265,1296,1268)(1270,1299,1327)(1273,1301,1324)(1276,1304,1321)(1279,1307,1318)(1282,1310,1315)(1285,1313,1312)(1288,1316,1309)(1291,1319,1306)(1294,1322,1303)(1297,1325,1300)(1302,1328,1353)(1305,1330,1350)(1308,1333,1347)(1311,1336,1344)(1314,1339,1341)(1317,1342,1338)(1320,1345,1335)(1323,1348,1332)(1326,1351,1329)(1331,1354,1376)(1334,1356,1373)(1337,1359,1370)(1340,1362,1367)(1343,1365,1364)(1346,1368,1361)(1349,1371,1358)(1352,1374,1355)(1357,1377,1396)(1360,1379,1393)(1363,1382,1390)(1366,1385,1387)(1369,1388,1384)(1372,1391,1381)(1375,1394,1378)(1380,1397,1413)(1383,1399,1410)(1386,1402,1407)(1389,1405,1404)(1392,1408,1401)(1395,1411,1398)(1400,1414,1427)(1403,1416,1424)(1406,1419,1421)(1409,1422,1418)(1412,1425,1415)(1417,1428,1438)(1420,1430,1435)(1423,1433,1432)(1426,1436,1429)(1431,1439,1446)(1434,1441,1443)(1437,1444,1440)(1442,1447,1451)(1445,1449,1448)(1450,1452,1453)(1455,1456);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,290,333,379,428,480,535,593,654,715,778,840,899,955,1008,1058,1105,1149,1190,1228,1263,1295,1324,1350,1373,1393,1410,1424,1435,1443,1448,1447,1430,1402,1362,1310,1246,1170,1082,982,870,746,619,503,399,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,385,434,486,541,599,660,723,785,841,713,590,648,709,772,834,893,949,1002,1052,1099,1143,1184,1222,1257,1289,1318,1344,1367,1387,1404,1418,1429,1428,1399,1359,1307,1243,1167,1079,979,867,743,616,500,396,304,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,302,345,391,440,492,547,605,666,729,791,850,906,956,838,710,588,477,529,584,642,703,766,828,887,943,996,1046,1093,1137,1178,1216,1251,1283,1312,1338,1361,1381,1398,1397,1356,1304,1240,1164,1076,976,864,740,613,497,393,301,221,153,97,53,71,91,113,138,166,197,231,268,308,351,397,446,498,553,611,672,735,797,856,912,965,1015,1059,953,835,707,585,475,376,422,471,523,578,636,697,760,822,881,937,990,1040,1087,1131,1172,1210,1245,1277,1306,1332,1355,1354,1301,1237,1161,1073,973,861,737,610,494,390,298,218,150,94,118,144,172,203,237,274,314,357,403,452,504,559,617,678,741,803,862,918,971,1021,1068,1112,1150,1056,950,832,704,582,472,374,287,327,370,416,465,517,572,630,691,754,816,875,931,984,1034,1081,1125,1166,1204,1239,1271,1300,1299,1234,1158,1070,970,858,734,607,491,387,295,215,147,177,209,243,280,320,363,409,458,510,565,623,684,747,809,868,924,977,1027,1074,1118,1159,1197,1229,1147,1053,947,829,701,579,469,371,285,210,244,281,321,364,410,459,511,566,624,685,748,810,869,925,978,1028,1075,1119,1160,1198,1233,1232,1155,1067,967,855,731,604,488,384,292,212,248,286,326,369,415,464,516,571,629,690,753,815,874,930,983,1033,1080,1124,1165,1203,1238,1270,1296,1226,1144,1050,944,826,698,576,466,368,282,208,145,173,204,238,275,315,358,404,453,505,560,618,679,742,804,863,919,972,1022,1069,1113,1154,1153,1064,964,852,728,601,485,381,289,331,375,421,470,522,577,635,696,759,821,880,936,989,1039,1086,1130,1171,1209,1244,1276,1305,1331,1351,1293,1223,1141,1047,941,823,695,573,463,365,279,205,143,92,114,139,167,198,232,269,309,352,398,447,499,554,612,673,736,798,857,913,966,1016,1063,1062,961,849,725,598,482,378,426,476,528,583,641,702,765,827,886,942,995,1045,1092,1136,1177,1215,1250,1282,1311,1337,1360,1380,1394,1348,1290,1220,1138,1044,938,820,692,570,460,362,276,202,140,90,51,67,86,108,133,161,192,226,263,303,346,392,441,493,548,606,667,730,792,851,907,960,959,846,722,595,479,533,589,647,708,771,833,892,948,1001,1051,1098,1142,1183,1221,1256,1288,1317,1343,1366,1386,1403,1417,1425,1391,1345,1287,1217,1135,1041,935,817,689,567,457,359,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,297,340,386,435,487,542,600,661,724,786,845,844,719,592,652,714,777,839,898,954,1007,1057,1104,1148,1189,1227,1262,1294,1323,1349,1372,1392,1409,1423,1434,1442,1444,1422,1388,1342,1284,1214,1132,1038,932,814,686,564,454,356,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,291,334,380,429,481,536,594,655,718,717,781,843,902,958,1011,1061,1108,1152,1193,1231,1266,1298,1327,1353,1376,1396,1413,1427,1438,1446,1451,1453,1452,1441,1419,1385,1339,1281,1211,1129,1035,929,811,683,561,451,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,382,431,483,538,596,657,720,779,651,712,775,837,896,952,1005,1055,1102,1146,1187,1225,1260,1292,1321,1347,1370,1390,1407,1421,1432,1440,1439,1416,1382,1336,1278,1208,1126,1032,926,808,680,558,448,350,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,342,388,437,489,544,602,663,726,788,847,900,776,649,532,587,645,706,769,831,890,946,999,1049,1096,1140,1181,1219,1254,1286,1315,1341,1364,1384,1401,1415,1414,1379,1333,1275,1205,1123,1029,923,805,677,555,445,347,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,348,394,443,495,550,608,669,732,794,853,909,962,1009,897,773,646,530,425,474,526,581,639,700,763,825,884,940,993,1043,1090,1134,1175,1213,1248,1280,1309,1335,1358,1378,1377,1330,1272,1202,1120,1026,920,802,674,552,442,344,258,184,122,72,93,116,141,169,200,234,271,311,354,400,449,501,556,614,675,738,800,859,915,968,1018,1065,1106,1006,894,770,643,527,423,330,373,419,468,520,575,633,694,757,819,878,934,987,1037,1084,1128,1169,1207,1242,1274,1303,1329,1328,1269,1199,1117,1023,917,799,671,549,439,341,255,181,119,146,175,206,240,277,317,360,406,455,507,562,620,681,744,806,865,921,974,1024,1071,1115,1156,1191,1103,1003,891,767,640,524,420,328,247,284,324,367,413,462,514,569,627,688,751,813,872,928,981,1031,1078,1122,1163,1201,1236,1268,1267,1196,1114,1020,914,796,668,546,436,338,252,178,211,246,283,323,366,412,461,513,568,626,687,750,812,871,927,980,1030,1077,1121,1162,1200,1235,1264,1188,1100,1000,888,764,637,521,417,325,245,176,207,241,278,318,361,407,456,508,563,621,682,745,807,866,922,975,1025,1072,1116,1157,1195,1194,1111,1017,911,793,665,543,433,335,249,288,329,372,418,467,519,574,632,693,756,818,877,933,986,1036,1083,1127,1168,1206,1241,1273,1302,1325,1261,1185,1097,997,885,761,634,518,414,322,242,174,117,142,170,201,235,272,312,355,401,450,502,557,615,676,739,801,860,916,969,1019,1066,1110,1109,1014,908,790,662,540,430,332,377,424,473,525,580,638,699,762,824,883,939,992,1042,1089,1133,1174,1212,1247,1279,1308,1334,1357,1374,1322,1258,1182,1094,994,882,758,631,515,411,319,239,171,115,70,89,111,136,164,195,229,266,306,349,395,444,496,551,609,670,733,795,854,910,963,1013,1012,905,787,659,537,427,478,531,586,644,705,768,830,889,945,998,1048,1095,1139,1180,1218,1253,1285,1314,1340,1363,1383,1400,1411,1371,1319,1255,1179,1091,991,879,755,628,512,408,316,236,168,112,68,35,48,64,83,105,130,158,189,223,260,300,343,389,438,490,545,603,664,727,789,848,904,903,784,656,534,591,650,711,774,836,895,951,1004,1054,1101,1145,1186,1224,1259,1291,1320,1346,1369,1389,1406,1420,1431,1436,1408,1368,1316,1252,1176,1088,988,876,752,625,509,405,313,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,294,337,383,432,484,539,597,658,721,783,782,653,716,780,842,901,957,1010,1060,1107,1151,1192,1230,1265,1297,1326,1352,1375,1395,1412,1426,1437,1445,1450,1449,1433,1405,1365,1313,1249,1173,1085,985,873,749,622,506,402,310,230,162,106,62,30,10)(1454,1455,1456)

Conjugacy classes & character table

1461 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26154, 10);

D3×C1453⋊C3 #39908526