D₃₀₆₈₉ #69849008

order	61378
automorphism	Aut(G) order 941784032
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₀₆₈₉

Splittings

Fitting, socle, derived, Sylow

C₃₀₆₈₉ ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃₀₆₈₉	C₂
LCS	G ▷ C₃₀₆₈₉
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30689
transitive degree	μ_t(G) = 30689

A minimal faithful permutation representation — 2 generators on 30689 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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838)(14854,15837)(14855,15836)(14856,15835)(14857,15834)(14858,15833)(14859,15832)(14860,15831)(14861,15830)(14862,15829)(14863,15828)(14864,15827)(14865,15826)(14866,15825)(14867,15824)(14868,15823)(14869,15822)(14870,15821)(14871,15820)(14872,15819)(14873,15818)(14874,15817)(14875,15816)(14876,15815)(14877,15814)(14878,15813)(14879,15812)(14880,15811)(14881,15810)(14882,15809)(14883,15808)(14884,15807)(14885,15806)(14886,15805)(14887,15804)(14888,15803)(14889,15802)(14890,15801)(14891,15800)(14892,15799)(14893,15798)(14894,15797)(14895,15796)(14896,15795)(14897,15794)(14898,15793)(14899,15792)(14900,15791)(14901,15790)(14902,15789)(14903,15788)(14904,15787)(14905,15786)(14906,15785)(14907,15784)(14908,15783)(14909,15782)(14910,15781)(14911,15780)(14912,15779)(14913,15778)(14914,15777)(14915,15776)(14916,15775)(14917,15774)(14918,15773)(14919,15772)(14920,15771)(14921,15770)(14922,15769)(14923,15768)(14924,15767)(14925,15766)(14926,15765)(14927,15764)(14928,15763)(14929,15762)(14930,15761)(14931,15760)(14932,15759)(14933,15758)(14934,15757)(14935,15756)(14936,15755)(14937,15754)(14938,15753)(14939,15752)(14940,15751)(14941,15750)(14942,15749)(14943,15748)(14944,15747)(14945,15746)(14946,15745)(14947,15744)(14948,15743)(14949,15742)(14950,15741)(14951,15740)(14952,15739)(14953,15738)(14954,15737)(14955,15736)(14956,15735)(14957,15734)(14958,15733)(14959,15732)(14960,15731)(14961,15730)(14962,15729)(14963,15728)(14964,15727)(14965,15726)(14966,15725)(14967,15724)(14968,15723)(14969,15722)(14970,15721)(14971,15720)(14972,15719)(14973,15718)(14974,15717)(14975,15716)(14976,15715)(14977,15714)(14978,15713)(14979,15712)(14980,15711)(14981,15710)(14982,15709)(14983,15708)(14984,15707)(14985,15706)(14986,15705)(14987,15704)(14988,15703)(14989,15702)(14990,15701)(14991,15700)(14992,15699)(14993,15698)(14994,15697)(14995,15696)(14996,15695)(14997,15694)(14998,15693)(14999,15692)(15000,15691)(15001,15690)(15002,15689)(15003,15688)(15004,15687)(15005,15686)(15006,15685)(15007,15684)(15008,15683)(15009,15682)(15010,15681)(15011,15680)(15012,15679)(15013,15678)(15014,15677)(15015,15676)(15016,15675)(15017,15674)(15018,15673)(15019,15672)(15020,15671)(15021,15670)(15022,15669)(15023,15668)(15024,15667)(15025,15666)(15026,15665)(15027,15664)(15028,15663)(15029,15662)(15030,15661)(15031,15660)(15032,15659)(15033,15658)(15034,15657)(15035,15656)(15036,15655)(15037,15654)(15038,15653)(15039,15652)(15040,15651)(15041,15650)(15042,15649)(15043,15648)(15044,15647)(15045,15646)(15046,15645)(15047,15644)(15048,15643)(15049,15642)(15050,15641)(15051,15640)(15052,15639)(15053,15638)(15054,15637)(15055,15636)(15056,15635)(15057,15634)(15058,15633)(15059,15632)(15060,15631)(15061,15630)(15062,15629)(15063,15628)(15064,15627)(15065,15626)(15066,15625)(15067,15624)(15068,15623)(15069,15622)(15070,15621)(15071,15620)(15072,15619)(15073,15618)(15074,15617)(15075,15616)(15076,15615)(15077,15614)(15078,15613)(15079,15612)(15080,15611)(15081,15610)(15082,15609)(15083,15608)(15084,15607)(15085,15606)(15086,15605)(15087,15604)(15088,15603)(15089,15602)(15090,15601)(15091,15600)(15092,15599)(15093,15598)(15094,15597)(15095,15596)(15096,15595)(15097,15594)(15098,15593)(15099,15592)(15100,15591)(15101,15590)(15102,15589)(15103,15588)(15104,15587)(15105,15586)(15106,15585)(15107,15584)(15108,15583)(15109,15582)(15110,15581)(15111,15580)(15112,15579)(15113,15578)(15114,15577)(15115,15576)(15116,15575)(15117,15574)(15118,15573)(15119,15572)(15120,15571)(15121,15570)(15122,15569)(15123,15568)(15124,15567)(15125,15566)(15126,15565)(15127,15564)(15128,15563)(15129,15562)(15130,15561)(15131,15560)(15132,15559)(15133,15558)(15134,15557)(15135,15556)(15136,15555)(15137,15554)(15138,15553)(15139,15552)(15140,15551)(15141,15550)(15142,15549)(15143,15548)(15144,15547)(15145,15546)(15146,15545)(15147,15544)(15148,15543)(15149,15542)(15150,15541)(15151,15540)(15152,15539)(15153,15538)(15154,15537)(15155,15536)(15156,15535)(15157,15534)(15158,15533)(15159,15532)(15160,15531)(15161,15530)(15162,15529)(15163,15528)(15164,15527)(15165,15526)(15166,15525)(15167,15524)(15168,15523)(15169,15522)(15170,15521)(15171,15520)(15172,15519)(15173,15518)(15174,15517)(15175,15516)(15176,15515)(15177,15514)(15178,15513)(15179,15512)(15180,15511)(15181,15510)(15182,15509)(15183,15508)(15184,15507)(15185,15506)(15186,15505)(15187,15504)(15188,15503)(15189,15502)(15190,15501)(15191,15500)(15192,15499)(15193,15498)(15194,15497)(15195,15496)(15196,15495)(15197,15494)(15198,15493)(15199,15492)(15200,15491)(15201,15490)(15202,15489)(15203,15488)(15204,15487)(15205,15486)(15206,15485)(15207,15484)(15208,15483)(15209,15482)(15210,15481)(15211,15480)(15212,15479)(15213,15478)(15214,15477)(15215,15476)(15216,15475)(15217,15474)(15218,15473)(15219,15472)(15220,15471)(15221,15470)(15222,15469)(15223,15468)(15224,15467)(15225,15466)(15226,15465)(15227,15464)(15228,15463)(15229,15462)(15230,15461)(15231,15460)(15232,15459)(15233,15458)(15234,15457)(15235,15456)(15236,15455)(15237,15454)(15238,15453)(15239,15452)(15240,15451)(15241,15450)(15242,15449)(15243,15448)(15244,15447)(15245,15446)(15246,15445)(15247,15444)(15248,15443)(15249,15442)(15250,15441)(15251,15440)(15252,15439)(15253,15438)(15254,15437)(15255,15436)(15256,15435)(15257,15434)(15258,15433)(15259,15432)(15260,15431)(15261,15430)(15262,15429)(15263,15428)(15264,15427)(15265,15426)(15266,15425)(15267,15424)(15268,15423)(15269,15422)(15270,15421)(15271,15420)(15272,15419)(15273,15418)(15274,15417)(15275,15416)(15276,15415)(15277,15414)(15278,15413)(15279,15412)(15280,15411)(15281,15410)(15282,15409)(15283,15408)(15284,15407)(15285,15406)(15286,15405)(15287,15404)(15288,15403)(15289,15402)(15290,15401)(15291,15400)(15292,15399)(15293,15398)(15294,15397)(15295,15396)(15296,15395)(15297,15394)(15298,15393)(15299,15392)(15300,15391)(15301,15390)(15302,15389)(15303,15388)(15304,15387)(15305,15386)(15306,15385)(15307,15384)(15308,15383)(15309,15382)(15310,15381)(15311,15380)(15312,15379)(15313,15378)(15314,15377)(15315,15376)(15316,15375)(15317,15374)(15318,15373)(15319,15372)(15320,15371)(15321,15370)(15322,15369)(15323,15368)(15324,15367)(15325,15366)(15326,15365)(15327,15364)(15328,15363)(15329,15362)(15330,15361)(15331,15360)(15332,15359)(15333,15358)(15334,15357)(15335,15356)(15336,15355)(15337,15354)(15338,15353)(15339,15352)(15340,15351)(15341,15350)(15342,15349)(15343,15348)(15344,15347)(15345,15346)

Polycyclic presentation:

2,30689|b^a=b^30688

Conjugacy classes & character table

15346 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(61378, 1);

D30689 #69849008