Finite Group DatabasebetaFinite Group Databasebeta| order | 23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(D1327⋊C3) |
| unambiguous semidirect product | (C1327⋊C3)⋊D3 |
| unambiguous semidirect product | D3981⋊C3 |
| Fitting, socle, derived | C3×C1327 ⋊ C2×C3 |
| Hall | C3×C1327⋊C3 ⋊ C2 |
| Sylow | C1327 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C1327 | C2×C3 |
| LCS | G ▷ C3×C1327 | |
| UCS | C1 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1330 |
| transitive degree | μt(G) = 3981 |
A minimal faithful permutation representation — 2 generators on 1330 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)(818,1021,1022,917,918,1129)(820,1019,1020,920,916,1013)(823,910,1018,922,923,1014)(830,1034,1035,929,930,1009)(837,1041,1042,936,937,1004)(844,1048,1049,943,944,999)(851,1055,1056,950,951,994)(858,1062,1063,957,958,989)(865,1069,1070,964,965,984)(871,1076,1077,971,972,979)(876,1083,1081,976,977,974)(881,1088,1074,981,982,968)(886,1093,1067,986,987,961)(891,1098,1060,991,992,954)(896,1103,1053,996,997,947)(901,1108,1046,1001,1002,940)(905,1113,1039,1006,1007,933)(908,1118,1032,1010,1011,926)(919,1134,1135,1024,1025,1246)(921,1132,1133,1027,1023,1126)(924,1017,1131,1029,1030,1127)(931,1146,1147,1036,1037,1122)(938,1153,1154,1043,1044,1117)(945,1160,1161,1050,1051,1112)(952,1167,1168,1057,1058,1107)(959,1174,1175,1064,1065,1102)(966,1181,1182,1071,1072,1097)(973,1188,1189,1078,1079,1092)(978,1195,1196,1084,1085,1087)(983,1201,1193,1089,1090,1082)(988,1206,1186,1094,1095,1075)(993,1211,1179,1099,1100,1068)(998,1216,1172,1104,1105,1061)(1003,1221,1165,1109,1110,1054)(1008,1226,1158,1114,1115,1047)(1012,1231,1151,1119,1120,1040)(1015,1236,1144,1123,1124,1033)(1026,1251,1252,1137,1138,1326)(1028,1249,1250,1140,1136,1244)(1031,1130,1248,1142,1139,1245)(1038,1257,1258,1148,1149,1240)(1045,1262,1263,1155,1156,1235)(1052,1267,1268,1162,1163,1230)(1059,1272,1273,1169,1170,1225)(1066,1277,1278,1176,1177,1220)(1073,1282,1283,1183,1184,1215)(1080,1287,1288,1190,1191,1210)(1086,1292,1293,1197,1198,1205)(1091,1297,1295,1202,1203,1200)(1096,1300,1290,1207,1208,1194)(1101,1303,1285,1212,1213,1187)(1106,1306,1280,1217,1218,1180)(1111,1309,1275,1222,1223,1173)(1116,1312,1270,1227,1228,1166)(1121,1315,1265,1232,1233,1159)(1125,1318,1260,1237,1238,1152)(1128,1321,1256,1241,1242,1145)(1141,1322,1261,1243,1253,1324)(1143,1247,1327,1254,1255,1325)(1150,1319,1266,1239,1259,1323)(1157,1316,1271,1234,1264,1320)(1164,1313,1276,1229,1269,1317)(1171,1310,1281,1224,1274,1314)(1178,1307,1286,1219,1279,1311)(1185,1304,1291,1214,1284,1308)(1192,1301,1296,1209,1289,1305)(1199,1298,1299,1204,1294,1302)(1329,1330);(1,2,5,11,19,32,51,74,103,138,179,226,279,338,403,474,551,634,723,818,919,1026,1139,918,1023,1037,1044,1051,1058,1065,1072,1079,1085,1090,1095,1100,1105,1110,1115,1120,1124,1127,1129,1244,1323,1265,1158,1046,939,838,743,654,571,494,423,358,299,246,195,147,97,47,66,82,67,46,65,92,124,161,203,251,305,364,429,500,577,660,749,844,945,1052,1164,1227,1114,1006,903,727,550,631,645,652,659,666,673,679,684,689,694,699,704,708,711,713,805,902,1004,1112,1225,1311,1285,1186,1074,967,866,767,671,581,497,419,347,273,175,218,258,253,248,241,234,219,174,217,268,324,385,451,523,601,685,775,871,973,1080,1192,1207,1094,986,884,788,698,614,535,407,278,335,349,356,363,369,374,379,384,388,391,393,461,534,612,696,786,882,984,1092,1205,1299,1288,1182,1063,949,841,739,643,545,399,466,530,525,520,515,510,503,496,489,482,467,398,465,540,620,705,795,891,993,1101,1214,1183,1071,964,863,768,678,594,516,444,378,318,263,183,102,135,149,155,160,164,167,169,213,262,316,376,442,514,592,676,765,860,961,1068,1180,1281,1268,1154,1035,913,719,810,898,893,888,883,878,873,868,861,854,847,840,833,826,811,718,809,908,1012,1121,1234,1155,1043,936,835,740,651,568,491,420,355,296,243,196,154,118,87,55,18,29,39,41,61,86,116,152,193,240,293,352,417,488,565,648,737,832,933,1040,1152,1261,1135,1248,1321,1318,1315,1312,1309,1306,1303,1300,1297,1292,1287,1282,1277,1272,1267,1262,1257,1249,1134,1247,1254,1142,1029,922,821,726,637,554,477,406,341,282,229,182,141,106,77,54,35,22,7,6,12,21,34,53,76,105,140,181,228,281,340,405,476,553,636,725,820,921,1028,1141,1241,1030,817,916,930,937,944,951,958,965,972,977,982,987,992,997,1002,1007,1011,1014,1016,1126,1240,1320,1270,1165,1053,946,845,750,661,578,501,430,365,302,242,188,132,70,95,117,111,96,69,94,127,165,208,256,310,370,436,507,584,667,756,851,952,1059,1171,1222,1109,1001,899,802,638,473,548,562,569,576,583,590,595,600,605,610,615,619,622,624,710,801,897,999,1107,1220,1308,1290,1193,1081,970,862,760,664,574,490,412,332,222,271,317,312,307,301,294,287,272,221,270,327,389,456,528,606,690,780,876,978,1086,1199,1202,1089,981,879,783,693,609,531,458,342,225,276,290,297,304,309,314,319,323,326,328,390,457,529,607,691,781,877,979,1087,1200,1296,1283,1175,1056,942,834,732,628,470,543,613,608,603,598,593,587,580,573,566,559,544,469,542,623,709,800,896,998,1106,1219,1176,1064,957,856,761,672,589,511,439,373,313,259,210,142,73,100,114,119,123,126,128,166,209,257,311,371,437,509,586,669,758,853,954,1061,1173,1276,1263,1147,1020,814,911,1005,1000,995,990,985,980,975,969,962,955,948,941,934,927,912,813,910,1015,1125,1239,1148,1036,929,828,733,644,561,484,413,348,289,236,189,148,113,83,58,36,10,16,24,38,57,81,110,145,186,233,286,345,410,481,558,641,730,825,926,1033,1145,1252,1327,1138,1253,1259,1264,1269,1274,1279,1284,1289,1294,1298,1301,1304,1307,1310,1313,1316,1319,1322,1251,1325,1326,1256,1144,1032,925,824,729,640,557,480,409,344,285,232,185,144,109,80,45,15,26,14,25,40,60,85,115,150,191,238,291,350,415,486,563,646,735,830,931,1038,1150,1237,1123,923,722,815,829,836,843,850,857,864,870,875,880,885,890,895,900,904,907,909,1013,1122,1235,1317,1275,1172,1060,953,852,757,668,585,508,433,361,295,235,173,99,130,158,153,146,131,98,129,168,212,261,315,375,441,513,591,674,763,858,959,1066,1178,1217,1104,996,894,798,707,555,402,471,485,492,499,506,512,517,522,527,532,536,539,541,621,706,796,892,994,1102,1215,1305,1295,1196,1077,963,855,753,657,567,483,397,275,330,382,377,372,367,360,353,346,331,274,329,392,460,533,611,695,785,881,983,1091,1204,1197,1084,976,874,778,688,604,526,454,387,283,178,223,237,244,250,255,260,264,267,269,325,386,452,524,602,686,776,872,974,1082,1194,1291,1278,1168,1049,935,827,717,547,626,702,697,692,687,682,677,670,663,656,649,642,627,546,625,712,804,901,1003,1111,1224,1169,1057,950,849,754,665,582,505,434,368,308,254,206,163,107,50,71,84,88,91,93,125,162,204,252,306,366,431,502,579,662,751,846,947,1054,1166,1271,1258,1133,915,1018,1118,1113,1108,1103,1098,1093,1088,1083,1076,1069,1062,1055,1048,1041,1034,1019,914,1017,1128,1243,1140,1027,920,819,724,635,552,475,404,339,280,227,180,139,104,75,52,33,20,4,8,13,23,37,56,79,108,143,184,231,284,343,408,479,556,639,728,823,924,1031,1143,1255,1025,1136,1149,1156,1163,1170,1177,1184,1191,1198,1203,1208,1213,1218,1223,1228,1233,1238,1242,1245,1246,1324,1260,1151,1039,932,831,736,647,564,487,416,351,292,239,192,151,112,68,28,43,44,27,42,63,89,120,156,198,245,298,357,422,493,570,653,742,837,938,1045,1157,1232,1119,1010,822,633,720,734,741,748,755,762,769,774,779,784,789,794,799,803,806,808,906,1009,1117,1230,1314,1280,1179,1067,960,859,764,675,588,504,426,354,288,220,134,171,205,200,194,187,172,133,170,215,265,320,380,446,518,596,680,770,865,966,1073,1185,1212,1099,991,889,793,703,618,478,337,400,414,421,428,435,440,445,450,455,459,462,464,538,617,701,791,887,989,1097,1210,1302,1293,1189,1070,956,848,746,650,560,468,334,395,453,448,443,438,432,425,418,411,396,333,394,463,537,616,700,790,886,988,1096,1209,1190,1078,971,869,773,683,599,521,449,383,322,230,137,176,190,197,202,207,211,214,216,266,321,381,447,519,597,681,771,867,968,1075,1187,1286,1273,1161,1042,928,812,630,715,797,792,787,782,777,772,766,759,752,745,738,731,716,629,714,807,905,1008,1116,1229,1162,1050,943,842,747,658,575,498,427,362,303,249,201,159,122,78,31,48,59,62,64,90,121,157,199,247,300,359,424,495,572,655,744,839,940,1047,1159,1266,1250,1022,1131,1236,1231,1226,1221,1216,1211,1206,1201,1195,1188,1181,1174,1167,1160,1153,1146,1132,1021,1130,1137,1024,917,816,721,632,549,472,401,336,277,224,177,136,101,72,49,30,17,9,3)(1328,1329,1330)Polycyclic presentation:
2,3^2,1327|c^a=c^2,d^a=d^1326,d^b=d^979672 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23886, 9);