D₃₀₅₀₉ #68744666

order	61018
automorphism	Aut(G) order 930768572
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₀₅₀₉

Splittings

Fitting, socle, derived, Sylow

C₃₀₅₀₉ ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃₀₅₀₉	C₂
LCS	G ▷ C₃₀₅₀₉
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30509
transitive degree	μ_t(G) = 30509

A minimal faithful permutation representation — 2 generators on 30509 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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75)(14937,15574)(14938,15573)(14939,15572)(14940,15571)(14941,15570)(14942,15569)(14943,15568)(14944,15567)(14945,15566)(14946,15565)(14947,15564)(14948,15563)(14949,15562)(14950,15561)(14951,15560)(14952,15559)(14953,15558)(14954,15557)(14955,15556)(14956,15555)(14957,15554)(14958,15553)(14959,15552)(14960,15551)(14961,15550)(14962,15549)(14963,15548)(14964,15547)(14965,15546)(14966,15545)(14967,15544)(14968,15543)(14969,15542)(14970,15541)(14971,15540)(14972,15539)(14973,15538)(14974,15537)(14975,15536)(14976,15535)(14977,15534)(14978,15533)(14979,15532)(14980,15531)(14981,15530)(14982,15529)(14983,15528)(14984,15527)(14985,15526)(14986,15525)(14987,15524)(14988,15523)(14989,15522)(14990,15521)(14991,15520)(14992,15519)(14993,15518)(14994,15517)(14995,15516)(14996,15515)(14997,15514)(14998,15513)(14999,15512)(15000,15511)(15001,15510)(15002,15509)(15003,15508)(15004,15507)(15005,15506)(15006,15505)(15007,15504)(15008,15503)(15009,15502)(15010,15501)(15011,15500)(15012,15499)(15013,15498)(15014,15497)(15015,15496)(15016,15495)(15017,15494)(15018,15493)(15019,15492)(15020,15491)(15021,15490)(15022,15489)(15023,15488)(15024,15487)(15025,15486)(15026,15485)(15027,15484)(15028,15483)(15029,15482)(15030,15481)(15031,15480)(15032,15479)(15033,15478)(15034,15477)(15035,15476)(15036,15475)(15037,15474)(15038,15473)(15039,15472)(15040,15471)(15041,15470)(15042,15469)(15043,15468)(15044,15467)(15045,15466)(15046,15465)(15047,15464)(15048,15463)(15049,15462)(15050,15461)(15051,15460)(15052,15459)(15053,15458)(15054,15457)(15055,15456)(15056,15455)(15057,15454)(15058,15453)(15059,15452)(15060,15451)(15061,15450)(15062,15449)(15063,15448)(15064,15447)(15065,15446)(15066,15445)(15067,15444)(15068,15443)(15069,15442)(15070,15441)(15071,15440)(15072,15439)(15073,15438)(15074,15437)(15075,15436)(15076,15435)(15077,15434)(15078,15433)(15079,15432)(15080,15431)(15081,15430)(15082,15429)(15083,15428)(15084,15427)(15085,15426)(15086,15425)(15087,15424)(15088,15423)(15089,15422)(15090,15421)(15091,15420)(15092,15419)(15093,15418)(15094,15417)(15095,15416)(15096,15415)(15097,15414)(15098,15413)(15099,15412)(15100,15411)(15101,15410)(15102,15409)(15103,15408)(15104,15407)(15105,15406)(15106,15405)(15107,15404)(15108,15403)(15109,15402)(15110,15401)(15111,15400)(15112,15399)(15113,15398)(15114,15397)(15115,15396)(15116,15395)(15117,15394)(15118,15393)(15119,15392)(15120,15391)(15121,15390)(15122,15389)(15123,15388)(15124,15387)(15125,15386)(15126,15385)(15127,15384)(15128,15383)(15129,15382)(15130,15381)(15131,15380)(15132,15379)(15133,15378)(15134,15377)(15135,15376)(15136,15375)(15137,15374)(15138,15373)(15139,15372)(15140,15371)(15141,15370)(15142,15369)(15143,15368)(15144,15367)(15145,15366)(15146,15365)(15147,15364)(15148,15363)(15149,15362)(15150,15361)(15151,15360)(15152,15359)(15153,15358)(15154,15357)(15155,15356)(15156,15355)(15157,15354)(15158,15353)(15159,15352)(15160,15351)(15161,15350)(15162,15349)(15163,15348)(15164,15347)(15165,15346)(15166,15345)(15167,15344)(15168,15343)(15169,15342)(15170,15341)(15171,15340)(15172,15339)(15173,15338)(15174,15337)(15175,15336)(15176,15335)(15177,15334)(15178,15333)(15179,15332)(15180,15331)(15181,15330)(15182,15329)(15183,15328)(15184,15327)(15185,15326)(15186,15325)(15187,15324)(15188,15323)(15189,15322)(15190,15321)(15191,15320)(15192,15319)(15193,15318)(15194,15317)(15195,15316)(15196,15315)(15197,15314)(15198,15313)(15199,15312)(15200,15311)(15201,15310)(15202,15309)(15203,15308)(15204,15307)(15205,15306)(15206,15305)(15207,15304)(15208,15303)(15209,15302)(15210,15301)(15211,15300)(15212,15299)(15213,15298)(15214,15297)(15215,15296)(15216,15295)(15217,15294)(15218,15293)(15219,15292)(15220,15291)(15221,15290)(15222,15289)(15223,15288)(15224,15287)(15225,15286)(15226,15285)(15227,15284)(15228,15283)(15229,15282)(15230,15281)(15231,15280)(15232,15279)(15233,15278)(15234,15277)(15235,15276)(15236,15275)(15237,15274)(15238,15273)(15239,15272)(15240,15271)(15241,15270)(15242,15269)(15243,15268)(15244,15267)(15245,15266)(15246,15265)(15247,15264)(15248,15263)(15249,15262)(15250,15261)(15251,15260)(15252,15259)(15253,15258)(15254,15257)(15255,15256)

Polycyclic presentation:

2,30509|b^a=b^30508

Conjugacy classes & character table

15256 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(61018, 1);

D30509 #68744666