Finite Group Databasebeta

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D₈₆₁₇ #31771908

• metacyclic
• supersolvable
• metabelian
• solvable
• monomial
• stem
• Frobenius

Aliases

Splittings

Subgroups

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 1238 points:

(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238);(2,7)(3,6)(4,5)(9,1238)(10,1237)(11,1236)(12,1235)(13,1234)(14,1233)(15,1232)(16,1231)(17,1230)(18,1229)(19,1228)(20,1227)(21,1226)(22,1225)(23,1224)(24,1223)(25,1222)(26,1221)(27,1220)(28,1219)(29,1218)(30,1217)(31,1216)(32,1215)(33,1214)(34,1213)(35,1212)(36,1211)(37,1210)(38,1209)(39,1208)(40,1207)(41,1206)(42,1205)(43,1204)(44,1203)(45,1202)(46,1201)(47,1200)(48,1199)(49,1198)(50,1197)(51,1196)(52,1195)(53,1194)(54,1193)(55,1192)(56,1191)(57,1190)(58,1189)(59,1188)(60,1187)(61,1186)(62,1185)(63,1184)(64,1183)(65,1182)(66,1181)(67,1180)(68,1179)(69,1178)(70,1177)(71,1176)(72,1175)(73,1174)(74,1173)(75,1172)(76,1171)(77,1170)(78,1169)(79,1168)(80,1167)(81,1166)(82,1165)(83,1164)(84,1163)(85,1162)(86,1161)(87,1160)(88,1159)(89,1158)(90,1157)(91,1156)(92,1155)(93,1154)(94,1153)(95,1152)(96,1151)(97,1150)(98,1149)(99,1148)(100,1147)(101,1146)(102,1145)(103,1144)(104,1143)(105,1142)(106,1141)(107,1140)(108,1139)(109,1138)(110,1137)(111,1136)(112,1135)(113,1134)(114,1133)(115,1132)(116,1131)(117,1130)(118,1129)(119,1128)(120,1127)(121,1126)(122,1125)(123,1124)(124,1123)(125,1122)(126,1121)(127,1120)(128,1119)(129,1118)(130,1117)(131,1116)(132,1115)(133,1114)(134,1113)(135,1112)(136,1111)(137,1110)(138,1109)(139,1108)(140,1107)(141,1106)(142,1105)(143,1104)(144,1103)(145,1102)(146,1101)(147,1100)(148,1099)(149,1098)(150,1097)(151,1096)(152,1095)(153,1094)(154,1093)(155,1092)(156,1091)(157,1090)(158,1089)(159,1088)(160,1087)(161,1086)(162,1085)(163,1084)(164,1083)(165,1082)(166,1081)(167,1080)(168,1079)(169,1078)(170,1077)(171,1076)(172,1075)(173,1074)(174,1073)(175,1072)(176,1071)(177,1070)(178,1069)(179,1068)(180,1067)(181,1066)(182,1065)(183,1064)(184,1063)(185,1062)(186,1061)(187,1060)(188,1059)(189,1058)(190,1057)(191,1056)(192,1055)(193,1054)(194,1053)(195,1052)(196,1051)(197,1050)(198,1049)(199,1048)(200,1047)(201,1046)(202,1045)(203,1044)(204,1043)(205,1042)(206,1041)(207,1040)(208,1039)(209,1038)(210,1037)(211,1036)(212,1035)(213,1034)(214,1033)(215,1032)(216,1031)(217,1030)(218,1029)(219,1028)(220,1027)(221,1026)(222,1025)(223,1024)(224,1023)(225,1022)(226,1021)(227,1020)(228,1019)(229,1018)(230,1017)(231,1016)(232,1015)(233,1014)(234,1013)(235,1012)(236,1011)(237,1010)(238,1009)(239,1008)(240,1007)(241,1006)(242,1005)(243,1004)(244,1003)(245,1002)(246,1001)(247,1000)(248,999)(249,998)(250,997)(251,996)(252,995)(253,994)(254,993)(255,992)(256,991)(257,990)(258,989)(259,988)(260,987)(261,986)(262,985)(263,984)(264,983)(265,982)(266,981)(267,980)(268,979)(269,978)(270,977)(271,976)(272,975)(273,974)(274,973)(275,972)(276,971)(277,970)(278,969)(279,968)(280,967)(281,966)(282,965)(283,964)(284,963)(285,962)(286,961)(287,960)(288,959)(289,958)(290,957)(291,956)(292,955)(293,954)(294,953)(295,952)(296,951)(297,950)(298,949)(299,948)(300,947)(301,946)(302,945)(303,944)(304,943)(305,942)(306,941)(307,940)(308,939)(309,938)(310,937)(311,936)(312,935)(313,934)(314,933)(315,932)(316,931)(317,930)(318,929)(319,928)(320,927)(321,926)(322,925)(323,924)(324,923)(325,922)(326,921)(327,920)(328,919)(329,918)(330,917)(331,916)(332,915)(333,914)(334,913)(335,912)(336,911)(337,910)(338,909)(339,908)(340,907)(341,906)(342,905)(343,904)(344,903)(345,902)(346,901)(347,900)(348,899)(349,898)(350,897)(351,896)(352,895)(353,894)(354,893)(355,892)(356,891)(357,890)(358,889)(359,888)(360,887)(361,886)(362,885)(363,884)(364,883)(365,882)(366,881)(367,880)(368,879)(369,878)(370,877)(371,876)(372,875)(373,874)(374,873)(375,872)(376,871)(377,870)(378,869)(379,868)(380,867)(381,866)(382,865)(383,864)(384,863)(385,862)(386,861)(387,860)(388,859)(389,858)(390,857)(391,856)(392,855)(393,854)(394,853)(395,852)(396,851)(397,850)(398,849)(399,848)(400,847)(401,846)(402,845)(403,844)(404,843)(405,842)(406,841)(407,840)(408,839)(409,838)(410,837)(411,836)(412,835)(413,834)(414,833)(415,832)(416,831)(417,830)(418,829)(419,828)(420,827)(421,826)(422,825)(423,824)(424,823)(425,822)(426,821)(427,820)(428,819)(429,818)(430,817)(431,816)(432,815)(433,814)(434,813)(435,812)(436,811)(437,810)(438,809)(439,808)(440,807)(441,806)(442,805)(443,804)(444,803)(445,802)(446,801)(447,800)(448,799)(449,798)(450,797)(451,796)(452,795)(453,794)(454,793)(455,792)(456,791)(457,790)(458,789)(459,788)(460,787)(461,786)(462,785)(463,784)(464,783)(465,782)(466,781)(467,780)(468,779)(469,778)(470,777)(471,776)(472,775)(473,774)(474,773)(475,772)(476,771)(477,770)(478,769)(479,768)(480,767)(481,766)(482,765)(483,764)(484,763)(485,762)(486,761)(487,760)(488,759)(489,758)(490,757)(491,756)(492,755)(493,754)(494,753)(495,752)(496,751)(497,750)(498,749)(499,748)(500,747)(501,746)(502,745)(503,744)(504,743)(505,742)(506,741)(507,740)(508,739)(509,738)(510,737)(511,736)(512,735)(513,734)(514,733)(515,732)(516,731)(517,730)(518,729)(519,728)(520,727)(521,726)(522,725)(523,724)(524,723)(525,722)(526,721)(527,720)(528,719)(529,718)(530,717)(531,716)(532,715)(533,714)(534,713)(535,712)(536,711)(537,710)(538,709)(539,708)(540,707)(541,706)(542,705)(543,704)(544,703)(545,702)(546,701)(547,700)(548,699)(549,698)(550,697)(551,696)(552,695)(553,694)(554,693)(555,692)(556,691)(557,690)(558,689)(559,688)(560,687)(561,686)(562,685)(563,684)(564,683)(565,682)(566,681)(567,680)(568,679)(569,678)(570,677)(571,676)(572,675)(573,674)(574,673)(575,672)(576,671)(577,670)(578,669)(579,668)(580,667)(581,666)(582,665)(583,664)(584,663)(585,662)(586,661)(587,660)(588,659)(589,658)(590,657)(591,656)(592,655)(593,654)(594,653)(595,652)(596,651)(597,650)(598,649)(599,648)(600,647)(601,646)(602,645)(603,644)(604,643)(605,642)(606,641)(607,640)(608,639)(609,638)(610,637)(611,636)(612,635)(613,634)(614,633)(615,632)(616,631)(617,630)(618,629)(619,628)(620,627)(621,626)(622,625)(623,624)

Polycyclic presentation:

2,7,1231|b^a=b^6,c^a=c^1230

Conjugacy classes & character table

4310 conjugacy classes. The character table is too large to display.

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17234, 3);