Finite Group DatabasebetaFinite Group Databasebeta| order | 21834 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(D1213⋊C3) |
| unambiguous semidirect product | (C1213⋊C3)⋊D3 |
| unambiguous semidirect product | D3639⋊C3 |
| Fitting, socle, derived | C3×C1213 ⋊ C2×C3 |
| Hall | C3×C1213⋊C3 ⋊ C2 |
| Sylow | C1213 ⋊ C3×D3 |
| 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.
| minimal degree | μ(G) = 1216 |
| transitive degree | μt(G) = 3639 |
A minimal faithful permutation representation — 2 generators on 1216 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,93)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,82,80,58,59,57)(51,98,99,72,73,128)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,111,112,83,84,86)(63,117,109,87,88,81)(74,133,134,101,102,169)(76,131,132,104,100,125)(79,94,130,106,107,126)(85,146,147,113,114,121)(89,153,151,118,119,116)(92,158,144,122,123,110)(103,174,175,136,137,216)(105,172,173,139,135,166)(108,129,171,141,142,167)(115,187,188,148,149,162)(120,194,195,154,155,157)(124,200,192,159,160,152)(127,205,185,163,164,145)(138,221,222,177,178,269)(140,219,220,180,176,213)(143,170,218,182,183,214)(150,234,235,189,190,209)(156,241,242,196,197,204)(161,248,246,201,202,199)(165,253,239,206,207,193)(168,258,232,210,211,186)(179,274,275,224,225,328)(181,272,273,227,223,266)(184,217,271,229,230,267)(191,287,288,236,237,262)(198,294,295,243,244,257)(203,301,302,249,250,252)(208,307,299,254,255,247)(212,312,292,259,260,240)(215,317,285,263,264,233)(226,333,334,277,278,393)(228,331,332,280,276,325)(231,270,330,282,283,326)(238,346,347,289,290,321)(245,353,354,296,297,316)(251,360,361,303,304,311)(256,367,365,308,309,306)(261,372,358,313,314,300)(265,377,351,318,319,293)(268,382,344,322,323,286)(279,398,399,336,337,464)(281,396,397,339,335,390)(284,329,395,341,342,391)(291,411,412,348,349,386)(298,418,419,355,356,381)(305,425,426,362,363,376)(310,432,433,368,369,371)(315,438,430,373,374,366)(320,443,423,378,379,359)(324,448,416,383,384,352)(327,453,409,387,388,345)(338,469,470,401,402,541)(340,467,468,404,400,461)(343,394,466,406,407,462)(350,482,483,413,414,457)(357,489,490,420,421,452)(364,496,497,427,428,447)(370,503,504,434,435,442)(375,510,508,439,440,437)(380,515,501,444,445,431)(385,520,494,449,450,424)(389,525,487,454,455,417)(392,530,480,458,459,410)(403,546,547,472,473,624)(405,544,545,475,471,538)(408,465,543,477,478,539)(415,559,560,484,485,534)(422,566,567,491,492,529)(429,573,574,498,499,524)(436,580,581,505,506,519)(441,587,588,511,512,514)(446,593,585,516,517,509)(451,598,578,521,522,502)(456,603,571,526,527,495)(460,608,564,531,532,488)(463,613,557,535,536,481)(474,629,630,549,550,713)(476,627,628,552,548,621)(479,542,626,554,555,622)(486,642,643,561,562,617)(493,649,650,568,569,612)(500,656,657,575,576,607)(507,663,664,582,583,602)(513,670,671,589,590,597)(518,677,675,594,595,592)(523,682,668,599,600,586)(528,687,661,604,605,579)(533,692,654,609,610,572)(537,697,647,614,615,565)(540,702,640,618,619,558)(551,718,719,632,633,808)(553,716,717,635,631,710)(556,625,715,637,638,711)(563,731,732,644,645,706)(570,738,739,651,652,701)(577,745,746,658,659,696)(584,752,753,665,666,691)(591,759,760,672,673,686)(596,766,767,678,679,681)(601,772,764,683,684,676)(606,777,757,688,689,669)(611,782,750,693,694,662)(616,787,743,698,699,655)(620,792,736,703,704,648)(623,797,729,707,708,641)(634,813,814,721,722,909)(636,811,812,724,720,805)(639,714,810,726,727,806)(646,826,827,733,734,801)(653,833,834,740,741,796)(660,840,841,747,748,791)(667,847,848,754,755,786)(674,854,855,761,762,781)(680,861,862,768,769,776)(685,868,866,773,774,771)(690,873,859,778,779,765)(695,878,852,783,784,758)(700,883,845,788,789,751)(705,888,838,793,794,744)(709,893,831,798,799,737)(712,898,824,802,803,730)(723,914,915,816,817,1016)(725,912,913,819,815,906)(728,809,911,821,822,907)(735,927,928,828,829,902)(742,934,935,835,836,897)(749,941,942,842,843,892)(756,948,949,849,850,887)(763,955,956,856,857,882)(770,962,963,863,864,877)(775,969,970,869,870,872)(780,975,967,874,875,867)(785,980,960,879,880,860)(790,985,953,884,885,853)(795,990,946,889,890,846)(800,995,939,894,895,839)(804,1000,932,899,900,832)(807,1005,925,903,904,825)(818,1021,1022,917,918,1115)(820,1019,1020,920,916,1013)(823,910,1018,922,923,1014)(830,1032,1033,929,930,1009)(837,1039,1040,936,937,1004)(844,1046,1047,943,944,999)(851,1053,1054,950,951,994)(858,1060,1061,957,958,989)(865,1065,1066,964,965,984)(871,1070,1071,971,972,979)(876,1075,1073,976,977,974)(881,1078,1068,981,982,968)(886,1081,1063,986,987,961)(891,1084,1058,991,992,954)(896,1089,1051,996,997,947)(901,1094,1044,1001,1002,940)(905,1099,1037,1006,1007,933)(908,1104,1030,1010,1011,926)(919,1082,1069,988,1024,1148)(921,1118,1119,1025,1023,1112)(924,1017,1117,1027,1028,1113)(931,1124,1125,1034,1035,1108)(938,1129,1130,1041,1042,1103)(945,1134,1135,1048,1049,1098)(952,1139,1140,1055,1056,1093)(959,1141,1144,1057,1062,1088)(966,1136,1147,1050,1067,1083)(973,1131,1150,1043,1072,1080)(978,1126,1152,1036,1076,1077)(983,1120,1151,1026,1079,1074)(993,1153,1145,1085,1086,1064)(998,1154,1142,1090,1091,1059)(1003,1158,1137,1095,1096,1052)(1008,1162,1132,1100,1101,1045)(1012,1166,1127,1105,1106,1038)(1015,1170,1122,1109,1110,1031)(1029,1116,1149,1087,1121,1177)(1092,1189,1186,1155,1156,1146)(1097,1190,1184,1159,1160,1143)(1102,1191,1182,1163,1164,1138)(1107,1195,1180,1167,1168,1133)(1111,1199,1178,1171,1172,1128)(1114,1188,1157,1174,1175,1123)(1161,1203,1204,1179,1176,1187)(1165,1210,1207,1192,1193,1185)(1169,1211,1205,1196,1197,1183)(1173,1209,1194,1200,1201,1181)(1198,1212,1213,1206,1202,1208)(1215,1216);(1,2,5,11,19,32,51,74,103,138,179,226,279,338,403,474,551,634,723,818,919,986,884,788,698,614,535,407,278,335,349,356,363,369,374,379,384,388,391,393,461,534,612,696,786,882,984,1080,1125,1020,814,911,1005,1000,995,990,985,980,975,969,962,955,948,941,934,927,912,813,910,1015,1111,1173,1202,1213,1207,1184,1142,1058,953,852,757,668,585,508,433,361,295,235,173,99,130,158,153,146,131,98,129,168,212,261,315,375,441,513,591,674,763,858,959,1055,950,849,754,665,582,505,434,368,308,254,206,163,107,50,71,84,88,91,93,125,162,204,252,306,366,431,502,579,662,751,846,947,1052,1138,1183,1206,1196,1167,1105,1010,822,633,720,734,741,748,755,762,769,774,779,784,789,794,799,803,806,808,906,1009,1103,1147,1061,949,841,739,643,545,399,466,530,525,520,515,510,503,496,489,482,467,398,465,540,620,705,795,891,993,1087,1027,922,821,726,637,554,477,406,341,282,229,182,141,106,77,54,35,22,7,6,12,21,34,53,76,105,140,181,228,281,340,405,476,553,636,725,820,921,1026,981,879,783,693,609,531,458,342,225,276,290,297,304,309,314,319,323,326,328,390,457,529,607,691,781,877,979,1077,1119,915,1018,1104,1099,1094,1089,1084,1081,1078,1075,1070,1065,1060,1053,1046,1039,1032,1019,914,1017,1114,1176,1193,1197,1201,1204,1186,1145,1063,960,859,764,675,588,504,426,354,288,220,134,171,205,200,194,187,172,133,170,215,265,320,380,446,518,596,680,770,865,966,1048,943,842,747,658,575,498,427,362,303,249,201,159,122,78,31,48,59,62,64,90,121,157,199,247,300,359,424,495,572,655,744,839,940,1045,1133,1181,1192,1163,1100,1006,903,727,550,631,645,652,659,666,673,679,684,689,694,699,704,708,711,713,805,902,1004,1098,1144,1054,942,834,732,628,470,543,613,608,603,598,593,587,580,573,566,559,544,469,542,623,709,800,896,998,1092,1157,1122,1030,925,824,729,640,557,480,409,344,285,232,185,144,109,80,45,15,26,14,25,40,60,85,115,150,191,238,291,350,415,486,563,646,735,830,931,1036,976,874,778,688,604,526,454,387,283,178,223,237,244,250,255,260,264,267,269,325,386,452,524,602,686,776,872,974,1074,1022,1117,1170,1166,1162,1158,1154,1153,1024,1120,1126,1131,1136,1141,1139,1134,1129,1124,1118,1021,1116,1156,1160,1164,1168,1172,1175,1177,1148,1068,967,866,767,671,581,497,419,347,273,175,218,258,253,248,241,234,219,174,217,268,324,385,451,523,601,685,775,871,973,1041,936,835,740,651,568,491,420,355,296,243,196,154,118,87,55,18,29,39,41,61,86,116,152,193,240,293,352,417,488,565,648,737,832,933,1038,1128,1179,1159,1095,1001,899,802,638,473,548,562,569,576,583,590,595,600,605,610,615,619,622,624,710,801,897,999,1093,1140,1047,935,827,717,547,626,702,697,692,687,682,677,670,663,656,649,642,627,546,625,712,804,901,1003,1097,1161,1178,1127,1037,932,831,736,647,564,487,416,351,292,239,192,151,112,68,28,43,44,27,42,63,89,120,156,198,245,298,357,422,493,570,653,742,837,938,1043,971,869,773,683,599,521,449,383,322,230,137,176,190,197,202,207,211,214,216,266,321,381,447,519,597,681,771,867,968,1069,1149,1188,1199,1195,1191,1190,1189,1121,918,1023,1035,1042,1049,1056,1062,1067,1072,1076,1079,1082,1086,1091,1096,1101,1106,1110,1113,1115,1151,1073,970,862,760,664,574,490,412,332,222,271,317,312,307,301,294,287,272,221,270,327,389,456,528,606,690,780,876,978,1034,929,828,733,644,561,484,413,348,289,236,189,148,113,83,58,36,10,16,24,38,57,81,110,145,186,233,286,345,410,481,558,641,730,825,926,1031,1123,1155,1090,996,894,798,707,555,402,471,485,492,499,506,512,517,522,527,532,536,539,541,621,706,796,892,994,1088,1135,1040,928,812,630,715,797,792,787,782,777,772,766,759,752,745,738,731,716,629,714,807,905,1008,1102,1165,1194,1180,1132,1044,939,838,743,654,571,494,423,358,299,246,195,147,97,47,66,82,67,46,65,92,124,161,203,251,305,364,429,500,577,660,749,844,945,1050,964,863,768,678,594,516,444,378,318,263,183,102,135,149,155,160,164,167,169,213,262,316,376,442,514,592,676,765,860,961,1064,1146,1187,1209,1211,1210,1203,1174,1028,817,916,930,937,944,951,958,965,972,977,982,987,992,997,1002,1007,1011,1014,1016,1112,1152,1071,963,855,753,657,567,483,397,275,330,382,377,372,367,360,353,346,331,274,329,392,460,533,611,695,785,881,983,1025,920,819,724,635,552,475,404,339,280,227,180,139,104,75,52,33,20,4,8,13,23,37,56,79,108,143,184,231,284,343,408,479,556,639,728,823,924,1029,1085,991,889,793,703,618,478,337,400,414,421,428,435,440,445,450,455,459,462,464,538,617,701,791,887,989,1083,1130,1033,913,719,810,898,893,888,883,878,873,868,861,854,847,840,833,826,811,718,809,908,1012,1107,1169,1198,1205,1182,1137,1051,946,845,750,661,578,501,430,365,302,242,188,132,70,95,117,111,96,69,94,127,165,208,256,310,370,436,507,584,667,756,851,952,1057,957,856,761,672,589,511,439,373,313,259,210,142,73,100,114,119,123,126,128,166,209,257,311,371,437,509,586,669,758,853,954,1059,1143,1185,1208,1212,1200,1171,1109,923,722,815,829,836,843,850,857,864,870,875,880,885,890,895,900,904,907,909,1013,1108,1150,1066,956,848,746,650,560,468,334,395,453,448,443,438,432,425,418,411,396,333,394,463,537,616,700,790,886,988,917,816,721,632,549,472,401,336,277,224,177,136,101,72,49,30,17,9,3)(1214,1215,1216)Polycyclic presentation:
2,3^2,1213|c^a=c^2,d^a=d^1212,d^b=d^995615 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21834, 9);