Finite Group DatabasebetaFinite Group Databasebeta| order | 34972 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic8743 |
| unambiguous semidirect product | C7⋊Dic1249 |
| derived, Hall | C7×C1249 ⋊ C4 |
| Sylow | C1249 ⋊ Dic7 |
| Sylow | C7 ⋊ Dic1249 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D8743 |
| commutator, hypocenter | C7×C1249 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C7×C1249 | C2 |
| LCS | G ▷ C7×C1249 | |
| UCS | C1 ◁ C2 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1260 |
| transitive degree | μt(G) = 34972 |
A minimal faithful permutation representation — 2 generators on 1260 points:
(1,2)(3,4)(5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260);(1,3,2,4)(6,11)(7,10)(8,9)(13,1260)(14,1259)(15,1258)(16,1257)(17,1256)(18,1255)(19,1254)(20,1253)(21,1252)(22,1251)(23,1250)(24,1249)(25,1248)(26,1247)(27,1246)(28,1245)(29,1244)(30,1243)(31,1242)(32,1241)(33,1240)(34,1239)(35,1238)(36,1237)(37,1236)(38,1235)(39,1234)(40,1233)(41,1232)(42,1231)(43,1230)(44,1229)(45,1228)(46,1227)(47,1226)(48,1225)(49,1224)(50,1223)(51,1222)(52,1221)(53,1220)(54,1219)(55,1218)(56,1217)(57,1216)(58,1215)(59,1214)(60,1213)(61,1212)(62,1211)(63,1210)(64,1209)(65,1208)(66,1207)(67,1206)(68,1205)(69,1204)(70,1203)(71,1202)(72,1201)(73,1200)(74,1199)(75,1198)(76,1197)(77,1196)(78,1195)(79,1194)(80,1193)(81,1192)(82,1191)(83,1190)(84,1189)(85,1188)(86,1187)(87,1186)(88,1185)(89,1184)(90,1183)(91,1182)(92,1181)(93,1180)(94,1179)(95,1178)(96,1177)(97,1176)(98,1175)(99,1174)(100,1173)(101,1172)(102,1171)(103,1170)(104,1169)(105,1168)(106,1167)(107,1166)(108,1165)(109,1164)(110,1163)(111,1162)(112,1161)(113,1160)(114,1159)(115,1158)(116,1157)(117,1156)(118,1155)(119,1154)(120,1153)(121,1152)(122,1151)(123,1150)(124,1149)(125,1148)(126,1147)(127,1146)(128,1145)(129,1144)(130,1143)(131,1142)(132,1141)(133,1140)(134,1139)(135,1138)(136,1137)(137,1136)(138,1135)(139,1134)(140,1133)(141,1132)(142,1131)(143,1130)(144,1129)(145,1128)(146,1127)(147,1126)(148,1125)(149,1124)(150,1123)(151,1122)(152,1121)(153,1120)(154,1119)(155,1118)(156,1117)(157,1116)(158,1115)(159,1114)(160,1113)(161,1112)(162,1111)(163,1110)(164,1109)(165,1108)(166,1107)(167,1106)(168,1105)(169,1104)(170,1103)(171,1102)(172,1101)(173,1100)(174,1099)(175,1098)(176,1097)(177,1096)(178,1095)(179,1094)(180,1093)(181,1092)(182,1091)(183,1090)(184,1089)(185,1088)(186,1087)(187,1086)(188,1085)(189,1084)(190,1083)(191,1082)(192,1081)(193,1080)(194,1079)(195,1078)(196,1077)(197,1076)(198,1075)(199,1074)(200,1073)(201,1072)(202,1071)(203,1070)(204,1069)(205,1068)(206,1067)(207,1066)(208,1065)(209,1064)(210,1063)(211,1062)(212,1061)(213,1060)(214,1059)(215,1058)(216,1057)(217,1056)(218,1055)(219,1054)(220,1053)(221,1052)(222,1051)(223,1050)(224,1049)(225,1048)(226,1047)(227,1046)(228,1045)(229,1044)(230,1043)(231,1042)(232,1041)(233,1040)(234,1039)(235,1038)(236,1037)(237,1036)(238,1035)(239,1034)(240,1033)(241,1032)(242,1031)(243,1030)(244,1029)(245,1028)(246,1027)(247,1026)(248,1025)(249,1024)(250,1023)(251,1022)(252,1021)(253,1020)(254,1019)(255,1018)(256,1017)(257,1016)(258,1015)(259,1014)(260,1013)(261,1012)(262,1011)(263,1010)(264,1009)(265,1008)(266,1007)(267,1006)(268,1005)(269,1004)(270,1003)(271,1002)(272,1001)(273,1000)(274,999)(275,998)(276,997)(277,996)(278,995)(279,994)(280,993)(281,992)(282,991)(283,990)(284,989)(285,988)(286,987)(287,986)(288,985)(289,984)(290,983)(291,982)(292,981)(293,980)(294,979)(295,978)(296,977)(297,976)(298,975)(299,974)(300,973)(301,972)(302,971)(303,970)(304,969)(305,968)(306,967)(307,966)(308,965)(309,964)(310,963)(311,962)(312,961)(313,960)(314,959)(315,958)(316,957)(317,956)(318,955)(319,954)(320,953)(321,952)(322,951)(323,950)(324,949)(325,948)(326,947)(327,946)(328,945)(329,944)(330,943)(331,942)(332,941)(333,940)(334,939)(335,938)(336,937)(337,936)(338,935)(339,934)(340,933)(341,932)(342,931)(343,930)(344,929)(345,928)(346,927)(347,926)(348,925)(349,924)(350,923)(351,922)(352,921)(353,920)(354,919)(355,918)(356,917)(357,916)(358,915)(359,914)(360,913)(361,912)(362,911)(363,910)(364,909)(365,908)(366,907)(367,906)(368,905)(369,904)(370,903)(371,902)(372,901)(373,900)(374,899)(375,898)(376,897)(377,896)(378,895)(379,894)(380,893)(381,892)(382,891)(383,890)(384,889)(385,888)(386,887)(387,886)(388,885)(389,884)(390,883)(391,882)(392,881)(393,880)(394,879)(395,878)(396,877)(397,876)(398,875)(399,874)(400,873)(401,872)(402,871)(403,870)(404,869)(405,868)(406,867)(407,866)(408,865)(409,864)(410,863)(411,862)(412,861)(413,860)(414,859)(415,858)(416,857)(417,856)(418,855)(419,854)(420,853)(421,852)(422,851)(423,850)(424,849)(425,848)(426,847)(427,846)(428,845)(429,844)(430,843)(431,842)(432,841)(433,840)(434,839)(435,838)(436,837)(437,836)(438,835)(439,834)(440,833)(441,832)(442,831)(443,830)(444,829)(445,828)(446,827)(447,826)(448,825)(449,824)(450,823)(451,822)(452,821)(453,820)(454,819)(455,818)(456,817)(457,816)(458,815)(459,814)(460,813)(461,812)(462,811)(463,810)(464,809)(465,808)(466,807)(467,806)(468,805)(469,804)(470,803)(471,802)(472,801)(473,800)(474,799)(475,798)(476,797)(477,796)(478,795)(479,794)(480,793)(481,792)(482,791)(483,790)(484,789)(485,788)(486,787)(487,786)(488,785)(489,784)(490,783)(491,782)(492,781)(493,780)(494,779)(495,778)(496,777)(497,776)(498,775)(499,774)(500,773)(501,772)(502,771)(503,770)(504,769)(505,768)(506,767)(507,766)(508,765)(509,764)(510,763)(511,762)(512,761)(513,760)(514,759)(515,758)(516,757)(517,756)(518,755)(519,754)(520,753)(521,752)(522,751)(523,750)(524,749)(525,748)(526,747)(527,746)(528,745)(529,744)(530,743)(531,742)(532,741)(533,740)(534,739)(535,738)(536,737)(537,736)(538,735)(539,734)(540,733)(541,732)(542,731)(543,730)(544,729)(545,728)(546,727)(547,726)(548,725)(549,724)(550,723)(551,722)(552,721)(553,720)(554,719)(555,718)(556,717)(557,716)(558,715)(559,714)(560,713)(561,712)(562,711)(563,710)(564,709)(565,708)(566,707)(567,706)(568,705)(569,704)(570,703)(571,702)(572,701)(573,700)(574,699)(575,698)(576,697)(577,696)(578,695)(579,694)(580,693)(581,692)(582,691)(583,690)(584,689)(585,688)(586,687)(587,686)(588,685)(589,684)(590,683)(591,682)(592,681)(593,680)(594,679)(595,678)(596,677)(597,676)(598,675)(599,674)(600,673)(601,672)(602,671)(603,670)(604,669)(605,668)(606,667)(607,666)(608,665)(609,664)(610,663)(611,662)(612,661)(613,660)(614,659)(615,658)(616,657)(617,656)(618,655)(619,654)(620,653)(621,652)(622,651)(623,650)(624,649)(625,648)(626,647)(627,646)(628,645)(629,644)(630,643)(631,642)(632,641)(633,640)(634,639)(635,638)(636,637)Polycyclic presentation:
2,7,1249,2|a^2=d,b^a=b^6,c^a=c^12488746 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34972, 4);