D₃×C₁₂₁₃⋊C₃ #36345709

order	21834
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₁₂₁₃⋊C₃
unambiguous semidirect product	C₃⋊(C₂×C₁₂₁₃⋊C₃)
unambiguous semidirect product	(D₃×C₁₂₁₃)⋊C₃

Splittings

Fitting, socle, derived	C₃×C₁₂₁₃ ⋊ C₂×C₃
Hall	C₃×C₁₂₁₃⋊C₃ ⋊ C₂
Sylow	C₁₂₁₃ ⋊ C₃×D₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₂₁₃	C₂×C₃
LCS	G ▷ C₃×C₁₂₁₃
UCS	C₁

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1216
transitive degree	μ_t(G) = 3639

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

Conjugacy classes & character table

1221 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21834, 10);

D3×C1213⋊C3 #36345709