D₂₇₆₂₄ #72290945

order	55248
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₆₂₄

Splittings

Fitting	C₃×C₈×C₁₁₅₁ ⋊ C₂
Hall	C₃×C₁₁₅₁ ⋊ D₈
Sylow	C₁₁₅₁ ⋊ D₂₄
Sylow	C₃ ⋊ D₉₂₀₈

Subgroups

	subgroup	G/H
center	C₂	D₁₃₈₁₂
commutator, cosocle	C₃×C₄×C₁₁₅₁	C₂²
Frattini	C₄	C₂×D₃₄₅₃
Fitting	C₃×C₈×C₁₁₅₁	C₂
socle	C₂×C₃×C₁₁₅₁	D₄
hypercenter	C₈	D₃₄₅₃
hypocenter	C₃×C₁₁₅₁	D₈
LCS	G ▷ C₃×C₄×C₁₁₅₁ ▷ C₂×C₃×C₁₁₅₁ ▷ C₃×C₁₁₅₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1162
transitive degree	μ_t(G) = 27624

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

Polycyclic presentation:

2^4,3,1151|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^1150

Conjugacy classes & character table

13815 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, 1151]);;
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.6^1150);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27624 #72290945