Finite Group Databasebeta

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D₁₃₆₉ #14582290

• metacyclic
• supersolvable
• metabelian
• solvable
• monolithic
• monomial
• stem
• Frobenius
• ambivalent

Aliases

Splittings

Subgroups

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 1369 points:

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

Polycyclic presentation:

2,37^2|b^37=c,b^a=b^36c^36,c^a=c^36

Conjugacy classes & character table

686 conjugacy classes — view the full character table.

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2738, 1);