D₃₂₂₈₄ #72302003

order	64568
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₂₈₄

Splittings

Fitting	C₄×C₇×C₁₁₅₃ ⋊ C₂
Hall	C₇×C₁₁₅₃ ⋊ D₄
Sylow	C₁₁₅₃ ⋊ D₂₈
Sylow	C₇ ⋊ D₄₆₁₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₈₀₇₁
commutator, socle, cosocle	C₂×C₇×C₁₁₅₃	C₂²
Fitting	C₄×C₇×C₁₁₅₃	C₂
hypercenter	C₄	D₈₀₇₁
hypocenter	C₇×C₁₁₅₃	D₄
LCS	G ▷ C₂×C₇×C₁₁₅₃ ▷ C₇×C₁₁₅₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1164
transitive degree	μ_t(G) = 32284

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

Polycyclic presentation:

2^3,7,1153|b^2=c,b^a=bc,d^a=d^6,e^a=e^1152

Conjugacy classes & character table

16145 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 7, 1153]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^1152);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32284 #72302003