D₂₈₀₆₉ #64159765

order	56138
automorphism	Aut(G) order 787840692
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₈₀₆₉

Splittings

Fitting, socle, derived, Sylow

C₂₈₀₆₉ ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂₈₀₆₉	C₂
LCS	G ▷ C₂₈₀₆₉
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28069
transitive degree	μ_t(G) = 28069

A minimal faithful permutation representation — 2 generators on 28069 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1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755,14316)(13756,14315)(13757,14314)(13758,14313)(13759,14312)(13760,14311)(13761,14310)(13762,14309)(13763,14308)(13764,14307)(13765,14306)(13766,14305)(13767,14304)(13768,14303)(13769,14302)(13770,14301)(13771,14300)(13772,14299)(13773,14298)(13774,14297)(13775,14296)(13776,14295)(13777,14294)(13778,14293)(13779,14292)(13780,14291)(13781,14290)(13782,14289)(13783,14288)(13784,14287)(13785,14286)(13786,14285)(13787,14284)(13788,14283)(13789,14282)(13790,14281)(13791,14280)(13792,14279)(13793,14278)(13794,14277)(13795,14276)(13796,14275)(13797,14274)(13798,14273)(13799,14272)(13800,14271)(13801,14270)(13802,14269)(13803,14268)(13804,14267)(13805,14266)(13806,14265)(13807,14264)(13808,14263)(13809,14262)(13810,14261)(13811,14260)(13812,14259)(13813,14258)(13814,14257)(13815,14256)(13816,14255)(13817,14254)(13818,14253)(13819,14252)(13820,14251)(13821,14250)(13822,14249)(13823,14248)(13824,14247)(13825,14246)(13826,14245)(13827,14244)(13828,14243)(13829,14242)(13830,14241)(13831,14240)(13832,14239)(13833,14238)(13834,14237)(13835,14236)(13836,14235)(13837,14234)(13838,14233)(13839,14232)(13840,14231)(13841,14230)(13842,14229)(13843,14228)(13844,14227)(13845,14226)(13846,14225)(13847,14224)(13848,14223)(13849,14222)(13850,14221)(13851,14220)(13852,14219)(13853,14218)(13854,14217)(13855,14216)(13856,14215)(13857,14214)(13858,14213)(13859,14212)(13860,14211)(13861,14210)(13862,14209)(13863,14208)(13864,14207)(13865,14206)(13866,14205)(13867,14204)(13868,14203)(13869,14202)(13870,14201)(13871,14200)(13872,14199)(13873,14198)(13874,14197)(13875,14196)(13876,14195)(13877,14194)(13878,14193)(13879,14192)(13880,14191)(13881,14190)(13882,14189)(13883,14188)(13884,14187)(13885,14186)(13886,14185)(13887,14184)(13888,14183)(13889,14182)(13890,14181)(13891,14180)(13892,14179)(13893,14178)(13894,14177)(13895,14176)(13896,14175)(13897,14174)(13898,14173)(13899,14172)(13900,14171)(13901,14170)(13902,14169)(13903,14168)(13904,14167)(13905,14166)(13906,14165)(13907,14164)(13908,14163)(13909,14162)(13910,14161)(13911,14160)(13912,14159)(13913,14158)(13914,14157)(13915,14156)(13916,14155)(13917,14154)(13918,14153)(13919,14152)(13920,14151)(13921,14150)(13922,14149)(13923,14148)(13924,14147)(13925,14146)(13926,14145)(13927,14144)(13928,14143)(13929,14142)(13930,14141)(13931,14140)(13932,14139)(13933,14138)(13934,14137)(13935,14136)(13936,14135)(13937,14134)(13938,14133)(13939,14132)(13940,14131)(13941,14130)(13942,14129)(13943,14128)(13944,14127)(13945,14126)(13946,14125)(13947,14124)(13948,14123)(13949,14122)(13950,14121)(13951,14120)(13952,14119)(13953,14118)(13954,14117)(13955,14116)(13956,14115)(13957,14114)(13958,14113)(13959,14112)(13960,14111)(13961,14110)(13962,14109)(13963,14108)(13964,14107)(13965,14106)(13966,14105)(13967,14104)(13968,14103)(13969,14102)(13970,14101)(13971,14100)(13972,14099)(13973,14098)(13974,14097)(13975,14096)(13976,14095)(13977,14094)(13978,14093)(13979,14092)(13980,14091)(13981,14090)(13982,14089)(13983,14088)(13984,14087)(13985,14086)(13986,14085)(13987,14084)(13988,14083)(13989,14082)(13990,14081)(13991,14080)(13992,14079)(13993,14078)(13994,14077)(13995,14076)(13996,14075)(13997,14074)(13998,14073)(13999,14072)(14000,14071)(14001,14070)(14002,14069)(14003,14068)(14004,14067)(14005,14066)(14006,14065)(14007,14064)(14008,14063)(14009,14062)(14010,14061)(14011,14060)(14012,14059)(14013,14058)(14014,14057)(14015,14056)(14016,14055)(14017,14054)(14018,14053)(14019,14052)(14020,14051)(14021,14050)(14022,14049)(14023,14048)(14024,14047)(14025,14046)(14026,14045)(14027,14044)(14028,14043)(14029,14042)(14030,14041)(14031,14040)(14032,14039)(14033,14038)(14034,14037)(14035,14036)

Polycyclic presentation:

2,28069|b^a=b^28068

Conjugacy classes & character table

14036 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(56138, 1);

D28069 #64159765