D₁₀₃₇₇ #35209276

order	20754
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₃₇₇
unambiguous semidirect product	C₉⋊D₁₁₅₃
unambiguous semidirect product	C₁₁₅₃⋊D₉

Splittings

Fitting, derived, Hall	C₉×C₁₁₅₃ ⋊ C₂
Sylow	C₁₁₅₃ ⋊ D₉
Sylow	C₉ ⋊ D₁₁₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₁₅₃	C₂
Frattini	C₃	D₃₄₅₉
socle	C₃×C₁₁₅₃	D₃
LCS	G ▷ C₉×C₁₁₅₃
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1162
transitive degree	μ_t(G) = 10377

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);(2,9)(3,8)(4,7)(5,6)(11,1162)(12,1161)(13,1160)(14,1159)(15,1158)(16,1157)(17,1156)(18,1155)(19,1154)(20,1153)(21,1152)(22,1151)(23,1150)(24,1149)(25,1148)(26,1147)(27,1146)(28,1145)(29,1144)(30,1143)(31,1142)(32,1141)(33,1140)(34,1139)(35,1138)(36,1137)(37,1136)(38,1135)(39,1134)(40,1133)(41,1132)(42,1131)(43,1130)(44,1129)(45,1128)(46,1127)(47,1126)(48,1125)(49,1124)(50,1123)(51,1122)(52,1121)(53,1120)(54,1119)(55,1118)(56,1117)(57,1116)(58,1115)(59,1114)(60,1113)(61,1112)(62,1111)(63,1110)(64,1109)(65,1108)(66,1107)(67,1106)(68,1105)(69,1104)(70,1103)(71,1102)(72,1101)(73,1100)(74,1099)(75,1098)(76,1097)(77,1096)(78,1095)(79,1094)(80,1093)(81,1092)(82,1091)(83,1090)(84,1089)(85,1088)(86,1087)(87,1086)(88,1085)(89,1084)(90,1083)(91,1082)(92,1081)(93,1080)(94,1079)(95,1078)(96,1077)(97,1076)(98,1075)(99,1074)(100,1073)(101,1072)(102,1071)(103,1070)(104,1069)(105,1068)(106,1067)(107,1066)(108,1065)(109,1064)(110,1063)(111,1062)(112,1061)(113,1060)(114,1059)(115,1058)(116,1057)(117,1056)(118,1055)(119,1054)(120,1053)(121,1052)(122,1051)(123,1050)(124,1049)(125,1048)(126,1047)(127,1046)(128,1045)(129,1044)(130,1043)(131,1042)(132,1041)(133,1040)(134,1039)(135,1038)(136,1037)(137,1036)(138,1035)(139,1034)(140,1033)(141,1032)(142,1031)(143,1030)(144,1029)(145,1028)(146,1027)(147,1026)(148,1025)(149,1024)(150,1023)(151,1022)(152,1021)(153,1020)(154,1019)(155,1018)(156,1017)(157,1016)(158,1015)(159,1014)(160,1013)(161,1012)(162,1011)(163,1010)(164,1009)(165,1008)(166,1007)(167,1006)(168,1005)(169,1004)(170,1003)(171,1002)(172,1001)(173,1000)(174,999)(175,998)(176,997)(177,996)(178,995)(179,994)(180,993)(181,992)(182,991)(183,990)(184,989)(185,988)(186,987)(187,986)(188,985)(189,984)(190,983)(191,982)(192,981)(193,980)(194,979)(195,978)(196,977)(197,976)(198,975)(199,974)(200,973)(201,972)(202,971)(203,970)(204,969)(205,968)(206,967)(207,966)(208,965)(209,964)(210,963)(211,962)(212,961)(213,960)(214,959)(215,958)(216,957)(217,956)(218,955)(219,954)(220,953)(221,952)(222,951)(223,950)(224,949)(225,948)(226,947)(227,946)(228,945)(229,944)(230,943)(231,942)(232,941)(233,940)(234,939)(235,938)(236,937)(237,936)(238,935)(239,934)(240,933)(241,932)(242,931)(243,930)(244,929)(245,928)(246,927)(247,926)(248,925)(249,924)(250,923)(251,922)(252,921)(253,920)(254,919)(255,918)(256,917)(257,916)(258,915)(259,914)(260,913)(261,912)(262,911)(263,910)(264,909)(265,908)(266,907)(267,906)(268,905)(269,904)(270,903)(271,902)(272,901)(273,900)(274,899)(275,898)(276,897)(277,896)(278,895)(279,894)(280,893)(281,892)(282,891)(283,890)(284,889)(285,888)(286,887)(287,886)(288,885)(289,884)(290,883)(291,882)(292,881)(293,880)(294,879)(295,878)(296,877)(297,876)(298,875)(299,874)(300,873)(301,872)(302,871)(303,870)(304,869)(305,868)(306,867)(307,866)(308,865)(309,864)(310,863)(311,862)(312,861)(313,860)(314,859)(315,858)(316,857)(317,856)(318,855)(319,854)(320,853)(321,852)(322,851)(323,850)(324,849)(325,848)(326,847)(327,846)(328,845)(329,844)(330,843)(331,842)(332,841)(333,840)(334,839)(335,838)(336,837)(337,836)(338,835)(339,834)(340,833)(341,832)(342,831)(343,830)(344,829)(345,828)(346,827)(347,826)(348,825)(349,824)(350,823)(351,822)(352,821)(353,820)(354,819)(355,818)(356,817)(357,816)(358,815)(359,814)(360,813)(361,812)(362,811)(363,810)(364,809)(365,808)(366,807)(367,806)(368,805)(369,804)(370,803)(371,802)(372,801)(373,800)(374,799)(375,798)(376,797)(377,796)(378,795)(379,794)(380,793)(381,792)(382,791)(383,790)(384,789)(385,788)(386,787)(387,786)(388,785)(389,784)(390,783)(391,782)(392,781)(393,780)(394,779)(395,778)(396,777)(397,776)(398,775)(399,774)(400,773)(401,772)(402,771)(403,770)(404,769)(405,768)(406,767)(407,766)(408,765)(409,764)(410,763)(411,762)(412,761)(413,760)(414,759)(415,758)(416,757)(417,756)(418,755)(419,754)(420,753)(421,752)(422,751)(423,750)(424,749)(425,748)(426,747)(427,746)(428,745)(429,744)(430,743)(431,742)(432,741)(433,740)(434,739)(435,738)(436,737)(437,736)(438,735)(439,734)(440,733)(441,732)(442,731)(443,730)(444,729)(445,728)(446,727)(447,726)(448,725)(449,724)(450,723)(451,722)(452,721)(453,720)(454,719)(455,718)(456,717)(457,716)(458,715)(459,714)(460,713)(461,712)(462,711)(463,710)(464,709)(465,708)(466,707)(467,706)(468,705)(469,704)(470,703)(471,702)(472,701)(473,700)(474,699)(475,698)(476,697)(477,696)(478,695)(479,694)(480,693)(481,692)(482,691)(483,690)(484,689)(485,688)(486,687)(487,686)(488,685)(489,684)(490,683)(491,682)(492,681)(493,680)(494,679)(495,678)(496,677)(497,676)(498,675)(499,674)(500,673)(501,672)(502,671)(503,670)(504,669)(505,668)(506,667)(507,666)(508,665)(509,664)(510,663)(511,662)(512,661)(513,660)(514,659)(515,658)(516,657)(517,656)(518,655)(519,654)(520,653)(521,652)(522,651)(523,650)(524,649)(525,648)(526,647)(527,646)(528,645)(529,644)(530,643)(531,642)(532,641)(533,640)(534,639)(535,638)(536,637)(537,636)(538,635)(539,634)(540,633)(541,632)(542,631)(543,630)(544,629)(545,628)(546,627)(547,626)(548,625)(549,624)(550,623)(551,622)(552,621)(553,620)(554,619)(555,618)(556,617)(557,616)(558,615)(559,614)(560,613)(561,612)(562,611)(563,610)(564,609)(565,608)(566,607)(567,606)(568,605)(569,604)(570,603)(571,602)(572,601)(573,600)(574,599)(575,598)(576,597)(577,596)(578,595)(579,594)(580,593)(581,592)(582,591)(583,590)(584,589)(585,588)(586,587)

Polycyclic presentation:

2,3,1153,3|b^3=d^2,b^a=b^2d,c^a=c^1152,d^a=d^2

Conjugacy classes & character table

5190 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁₁₃₁
maximal quotient of	D₃₁₁₃₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20754, 6);

D10377 #35209276