C₃×D₆₀₈₅ #49141010

order	36510
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₆₀₈₅
unambiguous semidirect product	C₅⋊(C₃×D₁₂₁₇)
unambiguous semidirect product	C₁₂₁₇⋊(C₃×D₅)

Splittings

Fitting, socle, Hall	C₃×C₅×C₁₂₁₇ ⋊ C₂
derived, Hall	C₅×C₁₂₁₇ ⋊ C₂×C₃
Hall	C₃×C₁₂₁₇ ⋊ D₅
Sylow	C₁₂₁₇ ⋊ C₃×D₅
Hall	C₃×C₅ ⋊ D₁₂₁₇
Sylow	C₅ ⋊ C₃×D₁₂₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₆₀₈₅
commutator, cosocle, hypocenter	C₅×C₁₂₁₇	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₂₁₇	C₂
LCS	G ▷ C₅×C₁₂₁₇
UCS	C₁ ◁ C₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1225
transitive degree	μ_t(G) = 18255

A minimal faithful permutation representation — 2 generators on 1225 points:

(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225);(5,8)(6,7)(10,1225)(11,1224)(12,1223)(13,1222)(14,1221)(15,1220)(16,1219)(17,1218)(18,1217)(19,1216)(20,1215)(21,1214)(22,1213)(23,1212)(24,1211)(25,1210)(26,1209)(27,1208)(28,1207)(29,1206)(30,1205)(31,1204)(32,1203)(33,1202)(34,1201)(35,1200)(36,1199)(37,1198)(38,1197)(39,1196)(40,1195)(41,1194)(42,1193)(43,1192)(44,1191)(45,1190)(46,1189)(47,1188)(48,1187)(49,1186)(50,1185)(51,1184)(52,1183)(53,1182)(54,1181)(55,1180)(56,1179)(57,1178)(58,1177)(59,1176)(60,1175)(61,1174)(62,1173)(63,1172)(64,1171)(65,1170)(66,1169)(67,1168)(68,1167)(69,1166)(70,1165)(71,1164)(72,1163)(73,1162)(74,1161)(75,1160)(76,1159)(77,1158)(78,1157)(79,1156)(80,1155)(81,1154)(82,1153)(83,1152)(84,1151)(85,1150)(86,1149)(87,1148)(88,1147)(89,1146)(90,1145)(91,1144)(92,1143)(93,1142)(94,1141)(95,1140)(96,1139)(97,1138)(98,1137)(99,1136)(100,1135)(101,1134)(102,1133)(103,1132)(104,1131)(105,1130)(106,1129)(107,1128)(108,1127)(109,1126)(110,1125)(111,1124)(112,1123)(113,1122)(114,1121)(115,1120)(116,1119)(117,1118)(118,1117)(119,1116)(120,1115)(121,1114)(122,1113)(123,1112)(124,1111)(125,1110)(126,1109)(127,1108)(128,1107)(129,1106)(130,1105)(131,1104)(132,1103)(133,1102)(134,1101)(135,1100)(136,1099)(137,1098)(138,1097)(139,1096)(140,1095)(141,1094)(142,1093)(143,1092)(144,1091)(145,1090)(146,1089)(147,1088)(148,1087)(149,1086)(150,1085)(151,1084)(152,1083)(153,1082)(154,1081)(155,1080)(156,1079)(157,1078)(158,1077)(159,1076)(160,1075)(161,1074)(162,1073)(163,1072)(164,1071)(165,1070)(166,1069)(167,1068)(168,1067)(169,1066)(170,1065)(171,1064)(172,1063)(173,1062)(174,1061)(175,1060)(176,1059)(177,1058)(178,1057)(179,1056)(180,1055)(181,1054)(182,1053)(183,1052)(184,1051)(185,1050)(186,1049)(187,1048)(188,1047)(189,1046)(190,1045)(191,1044)(192,1043)(193,1042)(194,1041)(195,1040)(196,1039)(197,1038)(198,1037)(199,1036)(200,1035)(201,1034)(202,1033)(203,1032)(204,1031)(205,1030)(206,1029)(207,1028)(208,1027)(209,1026)(210,1025)(211,1024)(212,1023)(213,1022)(214,1021)(215,1020)(216,1019)(217,1018)(218,1017)(219,1016)(220,1015)(221,1014)(222,1013)(223,1012)(224,1011)(225,1010)(226,1009)(227,1008)(228,1007)(229,1006)(230,1005)(231,1004)(232,1003)(233,1002)(234,1001)(235,1000)(236,999)(237,998)(238,997)(239,996)(240,995)(241,994)(242,993)(243,992)(244,991)(245,990)(246,989)(247,988)(248,987)(249,986)(250,985)(251,984)(252,983)(253,982)(254,981)(255,980)(256,979)(257,978)(258,977)(259,976)(260,975)(261,974)(262,973)(263,972)(264,971)(265,970)(266,969)(267,968)(268,967)(269,966)(270,965)(271,964)(272,963)(273,962)(274,961)(275,960)(276,959)(277,958)(278,957)(279,956)(280,955)(281,954)(282,953)(283,952)(284,951)(285,950)(286,949)(287,948)(288,947)(289,946)(290,945)(291,944)(292,943)(293,942)(294,941)(295,940)(296,939)(297,938)(298,937)(299,936)(300,935)(301,934)(302,933)(303,932)(304,931)(305,930)(306,929)(307,928)(308,927)(309,926)(310,925)(311,924)(312,923)(313,922)(314,921)(315,920)(316,919)(317,918)(318,917)(319,916)(320,915)(321,914)(322,913)(323,912)(324,911)(325,910)(326,909)(327,908)(328,907)(329,906)(330,905)(331,904)(332,903)(333,902)(334,901)(335,900)(336,899)(337,898)(338,897)(339,896)(340,895)(341,894)(342,893)(343,892)(344,891)(345,890)(346,889)(347,888)(348,887)(349,886)(350,885)(351,884)(352,883)(353,882)(354,881)(355,880)(356,879)(357,878)(358,877)(359,876)(360,875)(361,874)(362,873)(363,872)(364,871)(365,870)(366,869)(367,868)(368,867)(369,866)(370,865)(371,864)(372,863)(373,862)(374,861)(375,860)(376,859)(377,858)(378,857)(379,856)(380,855)(381,854)(382,853)(383,852)(384,851)(385,850)(386,849)(387,848)(388,847)(389,846)(390,845)(391,844)(392,843)(393,842)(394,841)(395,840)(396,839)(397,838)(398,837)(399,836)(400,835)(401,834)(402,833)(403,832)(404,831)(405,830)(406,829)(407,828)(408,827)(409,826)(410,825)(411,824)(412,823)(413,822)(414,821)(415,820)(416,819)(417,818)(418,817)(419,816)(420,815)(421,814)(422,813)(423,812)(424,811)(425,810)(426,809)(427,808)(428,807)(429,806)(430,805)(431,804)(432,803)(433,802)(434,801)(435,800)(436,799)(437,798)(438,797)(439,796)(440,795)(441,794)(442,793)(443,792)(444,791)(445,790)(446,789)(447,788)(448,787)(449,786)(450,785)(451,784)(452,783)(453,782)(454,781)(455,780)(456,779)(457,778)(458,777)(459,776)(460,775)(461,774)(462,773)(463,772)(464,771)(465,770)(466,769)(467,768)(468,767)(469,766)(470,765)(471,764)(472,763)(473,762)(474,761)(475,760)(476,759)(477,758)(478,757)(479,756)(480,755)(481,754)(482,753)(483,752)(484,751)(485,750)(486,749)(487,748)(488,747)(489,746)(490,745)(491,744)(492,743)(493,742)(494,741)(495,740)(496,739)(497,738)(498,737)(499,736)(500,735)(501,734)(502,733)(503,732)(504,731)(505,730)(506,729)(507,728)(508,727)(509,726)(510,725)(511,724)(512,723)(513,722)(514,721)(515,720)(516,719)(517,718)(518,717)(519,716)(520,715)(521,714)(522,713)(523,712)(524,711)(525,710)(526,709)(527,708)(528,707)(529,706)(530,705)(531,704)(532,703)(533,702)(534,701)(535,700)(536,699)(537,698)(538,697)(539,696)(540,695)(541,694)(542,693)(543,692)(544,691)(545,690)(546,689)(547,688)(548,687)(549,686)(550,685)(551,684)(552,683)(553,682)(554,681)(555,680)(556,679)(557,678)(558,677)(559,676)(560,675)(561,674)(562,673)(563,672)(564,671)(565,670)(566,669)(567,668)(568,667)(569,666)(570,665)(571,664)(572,663)(573,662)(574,661)(575,660)(576,659)(577,658)(578,657)(579,656)(580,655)(581,654)(582,653)(583,652)(584,651)(585,650)(586,649)(587,648)(588,647)(589,646)(590,645)(591,644)(592,643)(593,642)(594,641)(595,640)(596,639)(597,638)(598,637)(599,636)(600,635)(601,634)(602,633)(603,632)(604,631)(605,630)(606,629)(607,628)(608,627)(609,626)(610,625)(611,624)(612,623)(613,622)(614,621)(615,620)(616,619)(617,618)

Conjugacy classes & character table

9132 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36510, 4);

C3×D6085 #49141010