D₂₃₂₆₀ #72280038

order	46520
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₃₂₆₀

Splittings

Fitting	C₄×C₅×C₁₁₆₃ ⋊ C₂
Hall	C₅×C₁₁₆₃ ⋊ D₄
Sylow	C₁₁₆₃ ⋊ D₂₀
Sylow	C₅ ⋊ D₄₆₅₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₈₁₅
commutator, socle, cosocle	C₂×C₅×C₁₁₆₃	C₂²
Fitting	C₄×C₅×C₁₁₆₃	C₂
hypercenter	C₄	D₅₈₁₅
hypocenter	C₅×C₁₁₆₃	D₄
LCS	G ▷ C₂×C₅×C₁₁₆₃ ▷ C₅×C₁₁₆₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1172
transitive degree	μ_t(G) = 23260

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

Polycyclic presentation:

2^3,5,1163|b^2=c,b^a=bc,d^a=d^4,e^a=e^1162

Conjugacy classes & character table

11633 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, 5, 1163]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^1162);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D23260 #72280038