D₃×C₁₂₉₁⋊C₃ #37551233

order	23238
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₁₂₉₁⋊C₃
unambiguous semidirect product	C₃⋊(C₂×C₁₂₉₁⋊C₃)
unambiguous semidirect product	(D₃×C₁₂₉₁)⋊C₃

Splittings

Fitting, socle, derived	C₃×C₁₂₉₁ ⋊ C₂×C₃
Hall	C₃×C₁₂₉₁⋊C₃ ⋊ C₂
Sylow	C₁₂₉₁ ⋊ C₃×D₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₂₉₁	C₂×C₃
LCS	G ▷ C₃×C₁₂₉₁
UCS	C₁

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1294
transitive degree	μ_t(G) = 3873

A minimal faithful permutation representation — 2 generators on 1294 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)(654,717,729)(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)(720,783,791)(723,785,839)(726,788,836)(732,793,831)(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)(768,829,795)(771,832,792)(774,834,790)(777,837,787)(780,840,784)(781,794,782)(786,844,849)(789,846,894)(797,852,887)(800,855,884)(803,858,881)(806,861,878)(809,864,875)(812,867,872)(815,870,869)(818,873,866)(821,876,863)(824,879,860)(827,882,857)(830,885,854)(833,888,851)(835,890,850)(838,892,848)(841,895,845)(842,853,843)(847,899,901)(856,905,937)(859,908,934)(862,911,931)(865,914,928)(868,917,925)(871,920,922)(874,923,919)(877,926,916)(880,929,913)(883,932,910)(886,935,907)(889,938,904)(891,940,903)(893,942,902)(896,944,900)(897,906,898)(909,952,981)(912,955,978)(915,958,975)(918,961,972)(921,964,969)(924,967,966)(927,970,963)(930,973,960)(933,976,957)(936,979,954)(939,982,951)(941,984,950)(943,986,949)(945,988,948)(946,953,947)(956,996,1022)(959,999,1019)(962,1002,1016)(965,1005,1013)(968,1008,1010)(971,1011,1007)(974,1014,1004)(977,1017,1001)(980,1020,998)(983,1023,995)(985,1025,994)(987,1027,993)(989,1029,992)(990,997,991)(1000,1037,1060)(1003,1040,1057)(1006,1043,1054)(1009,1046,1051)(1012,1049,1048)(1015,1052,1045)(1018,1055,1042)(1021,1058,1039)(1024,1061,1036)(1026,1063,1035)(1028,1065,1034)(1030,1067,1033)(1031,1038,1032)(1041,1075,1095)(1044,1078,1092)(1047,1081,1089)(1050,1084,1086)(1053,1087,1083)(1056,1090,1080)(1059,1093,1077)(1062,1096,1074)(1064,1098,1073)(1066,1100,1072)(1068,1102,1071)(1069,1076,1070)(1079,1110,1127)(1082,1113,1124)(1085,1116,1121)(1088,1119,1118)(1091,1122,1115)(1094,1125,1112)(1097,1128,1109)(1099,1130,1108)(1101,1132,1107)(1103,1134,1106)(1104,1111,1105)(1114,1142,1156)(1117,1145,1153)(1120,1148,1150)(1123,1151,1147)(1126,1154,1144)(1129,1157,1141)(1131,1159,1140)(1133,1161,1139)(1135,1163,1138)(1136,1143,1137)(1146,1171,1182)(1149,1174,1179)(1152,1177,1176)(1155,1180,1173)(1158,1183,1170)(1160,1185,1169)(1162,1187,1168)(1164,1189,1167)(1165,1172,1166)(1175,1197,1205)(1178,1200,1202)(1181,1203,1199)(1184,1206,1196)(1186,1208,1195)(1188,1210,1194)(1190,1212,1193)(1191,1198,1192)(1201,1220,1225)(1204,1223,1222)(1207,1226,1219)(1209,1228,1218)(1211,1230,1217)(1213,1232,1216)(1214,1221,1215)(1224,1240,1242)(1227,1243,1239)(1229,1245,1238)(1231,1247,1237)(1233,1249,1236)(1234,1241,1235)(1244,1257,1256)(1246,1259,1255)(1248,1261,1254)(1250,1263,1253)(1251,1258,1252)(1260,1270,1269)(1262,1272,1268)(1264,1274,1267)(1265,1271,1266)(1273,1280,1279)(1275,1282,1278)(1276,1281,1277)(1283,1287,1286)(1284,1288,1285)(1289,1291,1290)(1293,1294);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,290,333,379,428,480,535,593,654,662,540,430,332,377,424,473,525,580,638,699,762,824,880,930,974,1015,1053,1088,1120,1149,1175,1198,1163,1100,1025,938,829,701,579,469,371,285,210,244,281,321,364,410,459,511,566,624,685,748,810,866,916,960,1001,1039,1074,1108,1139,1167,1192,1171,1113,1043,961,864,743,616,500,396,304,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,302,345,391,440,492,547,605,666,729,791,849,901,899,785,656,534,591,650,711,774,835,891,941,985,1026,1064,1099,1131,1160,1186,1209,1229,1246,1260,1271,1249,1210,1159,1096,1020,932,823,695,573,463,365,279,205,143,92,114,139,167,198,232,269,309,352,398,447,499,554,612,673,736,798,854,904,950,993,1033,1070,1037,955,858,737,610,494,390,298,218,150,94,118,144,172,203,237,274,314,357,403,452,504,559,617,678,741,803,859,909,953,895,776,649,532,587,645,706,769,831,887,937,981,1022,1060,1095,1127,1156,1182,1205,1225,1242,1256,1269,1279,1286,1290,1287,1272,1245,1206,1154,1090,1014,926,817,689,567,457,359,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,297,340,386,435,487,542,600,661,724,787,845,898,852,731,604,488,384,292,212,248,286,326,369,415,464,516,571,629,690,753,815,871,921,965,1006,1044,1079,1111,1067,986,890,770,643,527,423,330,373,419,468,520,575,633,694,757,819,875,925,969,1010,1048,1083,1115,1144,1170,1195,1217,1236,1252,1240,1200,1148,1084,1008,920,811,683,561,451,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,382,431,483,538,596,657,720,725,598,482,378,426,476,528,583,641,702,765,827,883,933,977,1018,1056,1091,1123,1152,1178,1201,1221,1189,1132,1063,982,885,764,637,521,417,325,245,176,207,241,278,318,361,407,456,508,563,621,682,745,807,863,913,957,998,1036,1073,1107,1138,1166,1142,1078,1002,914,805,677,555,445,347,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,348,394,443,495,550,608,669,732,794,715,778,839,894,844,719,592,652,714,777,838,893,943,987,1028,1066,1101,1133,1162,1188,1211,1231,1248,1262,1273,1281,1263,1230,1185,1128,1058,976,879,758,631,515,411,319,239,171,115,70,89,111,136,164,195,229,266,306,349,395,444,496,551,609,670,733,795,851,903,949,992,1032,996,908,799,671,549,439,341,255,181,119,146,175,206,240,277,317,360,406,455,507,562,620,681,744,806,862,912,956,997,944,837,710,588,477,529,584,642,703,766,828,884,934,978,1019,1057,1092,1124,1153,1179,1202,1222,1239,1255,1268,1278,1285,1280,1259,1226,1180,1122,1052,970,873,752,625,509,405,313,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,294,337,383,432,484,539,597,658,721,784,843,793,665,543,433,335,249,288,329,372,418,467,519,574,632,693,756,818,874,924,968,1009,1047,1082,1114,1143,1102,1027,940,832,704,582,472,374,287,327,370,416,465,517,572,630,691,754,816,872,922,966,1007,1045,1080,1112,1141,1169,1194,1216,1235,1220,1174,1116,1046,964,867,746,619,503,399,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,385,434,486,541,599,660,723,786,788,659,537,427,478,531,586,644,705,768,830,886,936,980,1021,1059,1094,1126,1155,1181,1204,1224,1241,1212,1161,1098,1023,935,826,698,576,466,368,282,208,145,173,204,238,275,315,358,404,453,505,560,618,679,742,804,860,910,954,995,1035,1072,1106,1137,1110,1040,958,861,740,613,497,393,301,221,153,97,53,71,91,113,138,166,197,231,268,308,351,397,446,498,553,611,672,735,797,853,779,651,712,775,836,783,653,716,780,841,896,945,989,1030,1068,1103,1135,1164,1190,1213,1233,1250,1264,1275,1283,1288,1274,1247,1208,1157,1093,1017,929,820,692,570,460,362,276,202,140,90,51,67,86,108,133,161,192,226,263,303,346,392,441,493,548,606,667,730,792,850,902,948,991,952,855,734,607,491,387,295,215,147,177,209,243,280,320,363,409,458,510,565,623,684,747,809,865,915,959,1000,1038,988,892,773,646,530,425,474,526,581,639,700,763,825,881,931,975,1016,1054,1089,1121,1150,1176,1199,1219,1238,1254,1267,1277,1270,1243,1203,1151,1087,1011,923,814,686,564,454,356,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,291,334,380,429,481,536,594,655,718,782,728,601,485,381,289,331,375,421,470,522,577,635,696,759,821,877,927,971,1012,1050,1085,1117,1146,1172,1134,1065,984,888,767,640,524,420,328,247,284,324,367,413,462,514,569,627,688,751,813,869,919,963,1004,1042,1077,1109,1140,1168,1193,1215,1197,1145,1081,1005,917,808,680,558,448,350,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,342,388,437,489,544,602,663,726,789,847,846,722,595,479,533,589,647,708,771,833,889,939,983,1024,1062,1097,1129,1158,1184,1207,1227,1244,1258,1232,1187,1130,1061,979,882,761,634,518,414,322,242,174,117,142,170,201,235,272,312,355,401,450,502,557,615,676,739,801,857,907,951,994,1034,1071,1105,1075,999,911,802,674,552,442,344,258,184,122,72,93,116,141,169,200,234,271,311,354,400,449,501,556,614,675,738,800,856,906,840,713,590,648,709,772,717,781,842,897,946,990,1031,1069,1104,1136,1165,1191,1214,1234,1251,1265,1276,1284,1289,1291,1282,1261,1228,1183,1125,1055,973,876,755,628,512,408,316,236,168,112,68,35,48,64,83,105,130,158,189,223,260,300,343,389,438,490,545,603,664,727,790,848,900,947,905,796,668,546,436,338,252,178,211,246,283,323,366,412,461,513,568,626,687,750,812,868,918,962,1003,1041,1076,1029,942,834,707,585,475,376,422,471,523,578,636,697,760,822,878,928,972,1013,1051,1086,1118,1147,1173,1196,1218,1237,1253,1266,1257,1223,1177,1119,1049,967,870,749,622,506,402,310,230,162,106,62,30,10)(1292,1293,1294)

Conjugacy classes & character table

1299 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23238, 10);

D3×C1291⋊C3 #37551233