D₂₄₀₂₀ #72285293

order	48040
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₄₀₂₀

Splittings

Fitting	C₄×C₅×C₁₂₀₁ ⋊ C₂
Hall	C₅×C₁₂₀₁ ⋊ D₄
Sylow	C₁₂₀₁ ⋊ D₂₀
Sylow	C₅ ⋊ D₄₈₀₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₀₀₅
commutator, socle, cosocle	C₂×C₅×C₁₂₀₁	C₂²
Fitting	C₄×C₅×C₁₂₀₁	C₂
hypercenter	C₄	D₆₀₀₅
hypocenter	C₅×C₁₂₀₁	D₄
LCS	G ▷ C₂×C₅×C₁₂₀₁ ▷ C₅×C₁₂₀₁
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1210
transitive degree	μ_t(G) = 24020

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

Polycyclic presentation:

2^3,5,1201|b^2=c,b^a=bc,d^a=d^4,e^a=e^1200

Conjugacy classes & character table

12013 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, 1201]);;
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.5^1200);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D24020 #72285293