Dic₁₅₈₂₈ #72295795

order	63312
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₈₂₈

Splittings

Hall	C₃×C₁₃₁₉ ⋊ Q₁₆
Sylow	C₁₃₁₉ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₅₂₇₆

Subgroups

	subgroup	G/H
center	C₂	D₁₅₈₂₈
commutator, cosocle	C₃×C₄×C₁₃₁₉	C₂²
Frattini	C₄	C₂×D₃₉₅₇
Fitting	C₃×C₈×C₁₃₁₉	C₂
socle, prime radical	C₂×C₃×C₁₃₁₉	D₄
hypercenter	C₈	D₃₉₅₇
hypocenter	C₃×C₁₃₁₉	Q₁₆
LCS	G ▷ C₃×C₄×C₁₃₁₉ ▷ C₂×C₃×C₁₃₁₉ ▷ C₃×C₁₃₁₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1338
transitive degree	μ_t(G) = 63312

A minimal faithful permutation representation — 2 generators on 1338 points:

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

Polycyclic presentation:

2^4,3,1319|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^1318

Conjugacy classes & character table

15831 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 1319]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^1318);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15828 #72295795