D₃×C₁₁₇₁⋊C₃ #35231120

order	21078
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₁₁₇₁⋊C₃
unambiguous semidirect product	C₃⋊(C₂×C₁₁₇₁⋊C₃)
unambiguous semidirect product	(D₃×C₁₁₇₁)⋊C₃

Splittings

Fitting, socle, derived	C₃×C₁₁₇₁ ⋊ C₂×C₃
Hall	C₃×C₁₁₇₁⋊C₃ ⋊ C₂
Sylow	C₁₁₇₁ ⋊ C₃×D₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₁₇₁	C₂×C₃
LCS	G ▷ C₃×C₁₁₇₁
UCS	C₁

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1174
transitive degree	μ_t(G) = 3513

A minimal faithful permutation representation — 2 generators on 1174 points:

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

Conjugacy classes & character table

1179 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21078, 12);

D3×C1171⋊C3 #35231120