D₃₀₁₀₉ #68736618

order	60218
automorphism	Aut(G) order 906521772
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₀₁₀₉

Splittings

Fitting, socle, derived, Sylow

C₃₀₁₀₉ ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃₀₁₀₉	C₂
LCS	G ▷ C₃₀₁₀₉
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30109
transitive degree	μ_t(G) = 30109

A minimal faithful permutation representation — 2 generators on 30109 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)(14737,15374)(14738,15373)(14739,15372)(14740,15371)(14741,15370)(14742,15369)(14743,15368)(14744,15367)(14745,15366)(14746,15365)(14747,15364)(14748,15363)(14749,15362)(14750,15361)(14751,15360)(14752,15359)(14753,15358)(14754,15357)(14755,15356)(14756,15355)(14757,15354)(14758,15353)(14759,15352)(14760,15351)(14761,15350)(14762,15349)(14763,15348)(14764,15347)(14765,15346)(14766,15345)(14767,15344)(14768,15343)(14769,15342)(14770,15341)(14771,15340)(14772,15339)(14773,15338)(14774,15337)(14775,15336)(14776,15335)(14777,15334)(14778,15333)(14779,15332)(14780,15331)(14781,15330)(14782,15329)(14783,15328)(14784,15327)(14785,15326)(14786,15325)(14787,15324)(14788,15323)(14789,15322)(14790,15321)(14791,15320)(14792,15319)(14793,15318)(14794,15317)(14795,15316)(14796,15315)(14797,15314)(14798,15313)(14799,15312)(14800,15311)(14801,15310)(14802,15309)(14803,15308)(14804,15307)(14805,15306)(14806,15305)(14807,15304)(14808,15303)(14809,15302)(14810,15301)(14811,15300)(14812,15299)(14813,15298)(14814,15297)(14815,15296)(14816,15295)(14817,15294)(14818,15293)(14819,15292)(14820,15291)(14821,15290)(14822,15289)(14823,15288)(14824,15287)(14825,15286)(14826,15285)(14827,15284)(14828,15283)(14829,15282)(14830,15281)(14831,15280)(14832,15279)(14833,15278)(14834,15277)(14835,15276)(14836,15275)(14837,15274)(14838,15273)(14839,15272)(14840,15271)(14841,15270)(14842,15269)(14843,15268)(14844,15267)(14845,15266)(14846,15265)(14847,15264)(14848,15263)(14849,15262)(14850,15261)(14851,15260)(14852,15259)(14853,15258)(14854,15257)(14855,15256)(14856,15255)(14857,15254)(14858,15253)(14859,15252)(14860,15251)(14861,15250)(14862,15249)(14863,15248)(14864,15247)(14865,15246)(14866,15245)(14867,15244)(14868,15243)(14869,15242)(14870,15241)(14871,15240)(14872,15239)(14873,15238)(14874,15237)(14875,15236)(14876,15235)(14877,15234)(14878,15233)(14879,15232)(14880,15231)(14881,15230)(14882,15229)(14883,15228)(14884,15227)(14885,15226)(14886,15225)(14887,15224)(14888,15223)(14889,15222)(14890,15221)(14891,15220)(14892,15219)(14893,15218)(14894,15217)(14895,15216)(14896,15215)(14897,15214)(14898,15213)(14899,15212)(14900,15211)(14901,15210)(14902,15209)(14903,15208)(14904,15207)(14905,15206)(14906,15205)(14907,15204)(14908,15203)(14909,15202)(14910,15201)(14911,15200)(14912,15199)(14913,15198)(14914,15197)(14915,15196)(14916,15195)(14917,15194)(14918,15193)(14919,15192)(14920,15191)(14921,15190)(14922,15189)(14923,15188)(14924,15187)(14925,15186)(14926,15185)(14927,15184)(14928,15183)(14929,15182)(14930,15181)(14931,15180)(14932,15179)(14933,15178)(14934,15177)(14935,15176)(14936,15175)(14937,15174)(14938,15173)(14939,15172)(14940,15171)(14941,15170)(14942,15169)(14943,15168)(14944,15167)(14945,15166)(14946,15165)(14947,15164)(14948,15163)(14949,15162)(14950,15161)(14951,15160)(14952,15159)(14953,15158)(14954,15157)(14955,15156)(14956,15155)(14957,15154)(14958,15153)(14959,15152)(14960,15151)(14961,15150)(14962,15149)(14963,15148)(14964,15147)(14965,15146)(14966,15145)(14967,15144)(14968,15143)(14969,15142)(14970,15141)(14971,15140)(14972,15139)(14973,15138)(14974,15137)(14975,15136)(14976,15135)(14977,15134)(14978,15133)(14979,15132)(14980,15131)(14981,15130)(14982,15129)(14983,15128)(14984,15127)(14985,15126)(14986,15125)(14987,15124)(14988,15123)(14989,15122)(14990,15121)(14991,15120)(14992,15119)(14993,15118)(14994,15117)(14995,15116)(14996,15115)(14997,15114)(14998,15113)(14999,15112)(15000,15111)(15001,15110)(15002,15109)(15003,15108)(15004,15107)(15005,15106)(15006,15105)(15007,15104)(15008,15103)(15009,15102)(15010,15101)(15011,15100)(15012,15099)(15013,15098)(15014,15097)(15015,15096)(15016,15095)(15017,15094)(15018,15093)(15019,15092)(15020,15091)(15021,15090)(15022,15089)(15023,15088)(15024,15087)(15025,15086)(15026,15085)(15027,15084)(15028,15083)(15029,15082)(15030,15081)(15031,15080)(15032,15079)(15033,15078)(15034,15077)(15035,15076)(15036,15075)(15037,15074)(15038,15073)(15039,15072)(15040,15071)(15041,15070)(15042,15069)(15043,15068)(15044,15067)(15045,15066)(15046,15065)(15047,15064)(15048,15063)(15049,15062)(15050,15061)(15051,15060)(15052,15059)(15053,15058)(15054,15057)(15055,15056)

Polycyclic presentation:

2,30109|b^a=b^30108

Conjugacy classes & character table

15056 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(60218, 1);

D30109 #68736618