D₂₇₉₅₁ #72291269

order	55902
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₇₉₅₁

Splittings

Fitting, derived, Hall	C₃×C₇×C₁₃₃₁ ⋊ C₂
Hall	C₇×C₁₃₃₁ ⋊ D₃
Hall	C₃×C₁₃₃₁ ⋊ D₇
Sylow	C₁₃₃₁ ⋊ D₂₁
Hall	C₃×C₇ ⋊ D₁₃₃₁
Sylow	C₇ ⋊ D₃₉₉₃
Sylow	C₃ ⋊ D₉₃₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₇×C₁₃₃₁	C₂
Frattini	C₁₂₁	D₂₃₁
socle	C₃×C₇×C₁₁	D₁₂₁
LCS	G ▷ C₃×C₇×C₁₃₃₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1341
transitive degree	μ_t(G) = 27951

A minimal faithful permutation representation — 2 generators on 1341 points:

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

Polycyclic presentation:

2,3,7,11^3|d^11=e,e^11=f,b^a=b^2,c^a=c^6,d^a=d^10e^10f^10,e^a=e^10f^10,f^a=f^10

Conjugacy classes & character table

13977 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 3, 7, 11, 11, 11]);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^10*F.5^10*F.6^10);;
gap> SetConjugate(coll, 5, 1, F.5^10*F.6^10);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27951 #72291269