Finite Group DatabasebetaFinite Group Databasebeta| order | 34468 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic8617 |
| unambiguous semidirect product | C7⋊Dic1231 |
| unambiguous semidirect product | C1231⋊Dic7 |
| derived, Hall | C7×C1231 ⋊ C4 |
| Sylow | C1231 ⋊ Dic7 |
| Sylow | C7 ⋊ Dic1231 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D8617 |
| commutator, hypocenter | C7×C1231 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C7×C1231 | C2 |
| LCS | G ▷ C7×C1231 | |
| UCS | C1 ◁ C2 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1242 |
| transitive degree | μt(G) = 34468 |
A minimal faithful permutation representation — 2 generators on 1242 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,1242)(14,1241)(15,1240)(16,1239)(17,1238)(18,1237)(19,1236)(20,1235)(21,1234)(22,1233)(23,1232)(24,1231)(25,1230)(26,1229)(27,1228)(28,1227)(29,1226)(30,1225)(31,1224)(32,1223)(33,1222)(34,1221)(35,1220)(36,1219)(37,1218)(38,1217)(39,1216)(40,1215)(41,1214)(42,1213)(43,1212)(44,1211)(45,1210)(46,1209)(47,1208)(48,1207)(49,1206)(50,1205)(51,1204)(52,1203)(53,1202)(54,1201)(55,1200)(56,1199)(57,1198)(58,1197)(59,1196)(60,1195)(61,1194)(62,1193)(63,1192)(64,1191)(65,1190)(66,1189)(67,1188)(68,1187)(69,1186)(70,1185)(71,1184)(72,1183)(73,1182)(74,1181)(75,1180)(76,1179)(77,1178)(78,1177)(79,1176)(80,1175)(81,1174)(82,1173)(83,1172)(84,1171)(85,1170)(86,1169)(87,1168)(88,1167)(89,1166)(90,1165)(91,1164)(92,1163)(93,1162)(94,1161)(95,1160)(96,1159)(97,1158)(98,1157)(99,1156)(100,1155)(101,1154)(102,1153)(103,1152)(104,1151)(105,1150)(106,1149)(107,1148)(108,1147)(109,1146)(110,1145)(111,1144)(112,1143)(113,1142)(114,1141)(115,1140)(116,1139)(117,1138)(118,1137)(119,1136)(120,1135)(121,1134)(122,1133)(123,1132)(124,1131)(125,1130)(126,1129)(127,1128)(128,1127)(129,1126)(130,1125)(131,1124)(132,1123)(133,1122)(134,1121)(135,1120)(136,1119)(137,1118)(138,1117)(139,1116)(140,1115)(141,1114)(142,1113)(143,1112)(144,1111)(145,1110)(146,1109)(147,1108)(148,1107)(149,1106)(150,1105)(151,1104)(152,1103)(153,1102)(154,1101)(155,1100)(156,1099)(157,1098)(158,1097)(159,1096)(160,1095)(161,1094)(162,1093)(163,1092)(164,1091)(165,1090)(166,1089)(167,1088)(168,1087)(169,1086)(170,1085)(171,1084)(172,1083)(173,1082)(174,1081)(175,1080)(176,1079)(177,1078)(178,1077)(179,1076)(180,1075)(181,1074)(182,1073)(183,1072)(184,1071)(185,1070)(186,1069)(187,1068)(188,1067)(189,1066)(190,1065)(191,1064)(192,1063)(193,1062)(194,1061)(195,1060)(196,1059)(197,1058)(198,1057)(199,1056)(200,1055)(201,1054)(202,1053)(203,1052)(204,1051)(205,1050)(206,1049)(207,1048)(208,1047)(209,1046)(210,1045)(211,1044)(212,1043)(213,1042)(214,1041)(215,1040)(216,1039)(217,1038)(218,1037)(219,1036)(220,1035)(221,1034)(222,1033)(223,1032)(224,1031)(225,1030)(226,1029)(227,1028)(228,1027)(229,1026)(230,1025)(231,1024)(232,1023)(233,1022)(234,1021)(235,1020)(236,1019)(237,1018)(238,1017)(239,1016)(240,1015)(241,1014)(242,1013)(243,1012)(244,1011)(245,1010)(246,1009)(247,1008)(248,1007)(249,1006)(250,1005)(251,1004)(252,1003)(253,1002)(254,1001)(255,1000)(256,999)(257,998)(258,997)(259,996)(260,995)(261,994)(262,993)(263,992)(264,991)(265,990)(266,989)(267,988)(268,987)(269,986)(270,985)(271,984)(272,983)(273,982)(274,981)(275,980)(276,979)(277,978)(278,977)(279,976)(280,975)(281,974)(282,973)(283,972)(284,971)(285,970)(286,969)(287,968)(288,967)(289,966)(290,965)(291,964)(292,963)(293,962)(294,961)(295,960)(296,959)(297,958)(298,957)(299,956)(300,955)(301,954)(302,953)(303,952)(304,951)(305,950)(306,949)(307,948)(308,947)(309,946)(310,945)(311,944)(312,943)(313,942)(314,941)(315,940)(316,939)(317,938)(318,937)(319,936)(320,935)(321,934)(322,933)(323,932)(324,931)(325,930)(326,929)(327,928)(328,927)(329,926)(330,925)(331,924)(332,923)(333,922)(334,921)(335,920)(336,919)(337,918)(338,917)(339,916)(340,915)(341,914)(342,913)(343,912)(344,911)(345,910)(346,909)(347,908)(348,907)(349,906)(350,905)(351,904)(352,903)(353,902)(354,901)(355,900)(356,899)(357,898)(358,897)(359,896)(360,895)(361,894)(362,893)(363,892)(364,891)(365,890)(366,889)(367,888)(368,887)(369,886)(370,885)(371,884)(372,883)(373,882)(374,881)(375,880)(376,879)(377,878)(378,877)(379,876)(380,875)(381,874)(382,873)(383,872)(384,871)(385,870)(386,869)(387,868)(388,867)(389,866)(390,865)(391,864)(392,863)(393,862)(394,861)(395,860)(396,859)(397,858)(398,857)(399,856)(400,855)(401,854)(402,853)(403,852)(404,851)(405,850)(406,849)(407,848)(408,847)(409,846)(410,845)(411,844)(412,843)(413,842)(414,841)(415,840)(416,839)(417,838)(418,837)(419,836)(420,835)(421,834)(422,833)(423,832)(424,831)(425,830)(426,829)(427,828)(428,827)(429,826)(430,825)(431,824)(432,823)(433,822)(434,821)(435,820)(436,819)(437,818)(438,817)(439,816)(440,815)(441,814)(442,813)(443,812)(444,811)(445,810)(446,809)(447,808)(448,807)(449,806)(450,805)(451,804)(452,803)(453,802)(454,801)(455,800)(456,799)(457,798)(458,797)(459,796)(460,795)(461,794)(462,793)(463,792)(464,791)(465,790)(466,789)(467,788)(468,787)(469,786)(470,785)(471,784)(472,783)(473,782)(474,781)(475,780)(476,779)(477,778)(478,777)(479,776)(480,775)(481,774)(482,773)(483,772)(484,771)(485,770)(486,769)(487,768)(488,767)(489,766)(490,765)(491,764)(492,763)(493,762)(494,761)(495,760)(496,759)(497,758)(498,757)(499,756)(500,755)(501,754)(502,753)(503,752)(504,751)(505,750)(506,749)(507,748)(508,747)(509,746)(510,745)(511,744)(512,743)(513,742)(514,741)(515,740)(516,739)(517,738)(518,737)(519,736)(520,735)(521,734)(522,733)(523,732)(524,731)(525,730)(526,729)(527,728)(528,727)(529,726)(530,725)(531,724)(532,723)(533,722)(534,721)(535,720)(536,719)(537,718)(538,717)(539,716)(540,715)(541,714)(542,713)(543,712)(544,711)(545,710)(546,709)(547,708)(548,707)(549,706)(550,705)(551,704)(552,703)(553,702)(554,701)(555,700)(556,699)(557,698)(558,697)(559,696)(560,695)(561,694)(562,693)(563,692)(564,691)(565,690)(566,689)(567,688)(568,687)(569,686)(570,685)(571,684)(572,683)(573,682)(574,681)(575,680)(576,679)(577,678)(578,677)(579,676)(580,675)(581,674)(582,673)(583,672)(584,671)(585,670)(586,669)(587,668)(588,667)(589,666)(590,665)(591,664)(592,663)(593,662)(594,661)(595,660)(596,659)(597,658)(598,657)(599,656)(600,655)(601,654)(602,653)(603,652)(604,651)(605,650)(606,649)(607,648)(608,647)(609,646)(610,645)(611,644)(612,643)(613,642)(614,641)(615,640)(616,639)(617,638)(618,637)(619,636)(620,635)(621,634)(622,633)(623,632)(624,631)(625,630)(626,629)(627,628)Polycyclic presentation:
2,7,1231,2|a^2=d,b^a=b^6,c^a=c^12308620 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34468, 4);