Finite Group Databasebeta

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Dic₄₇₇₂ #34061796

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

Aliases

Splittings

Subgroups

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 1209 points:

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

Polycyclic presentation:

2^4,1193|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^1192,c^b=cd,e^b=e^1192

Conjugacy classes & character table

4775 conjugacy classes. The character table is too large to display.

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19088, 31);