D₃×C₁₂₇₉⋊C₃ #37527722

order	23022
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₁₂₇₉⋊C₃
unambiguous semidirect product	C₃⋊(C₂×C₁₂₇₉⋊C₃)
unambiguous semidirect product	(D₃×C₁₂₇₉)⋊C₃

Splittings

Fitting, socle, derived	C₃×C₁₂₇₉ ⋊ C₂×C₃
Hall	C₃×C₁₂₇₉⋊C₃ ⋊ C₂
Sylow	C₁₂₇₉ ⋊ C₃×D₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₂₇₉	C₂×C₃
LCS	G ▷ C₃×C₁₂₇₉
UCS	C₁

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1282
transitive degree	μ_t(G) = 3837

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

Conjugacy classes & character table

1287 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23022, 12);

D3×C1279⋊C3 #37527722