Dic₁₃₈₃₆ #72290118

order	55344
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₈₃₆

Splittings

Hall	C₃×C₁₁₅₃ ⋊ Q₁₆
Sylow	C₁₁₅₃ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₄₆₁₂

Subgroups

	subgroup	G/H
center	C₂	D₁₃₈₃₆
commutator, cosocle	C₃×C₄×C₁₁₅₃	C₂²
Frattini	C₄	C₂×D₃₄₅₉
Fitting	C₃×C₈×C₁₁₅₃	C₂
socle, prime radical	C₂×C₃×C₁₁₅₃	D₄
hypercenter	C₈	D₃₄₅₉
hypocenter	C₃×C₁₁₅₃	Q₁₆
LCS	G ▷ C₃×C₄×C₁₁₅₃ ▷ C₂×C₃×C₁₁₅₃ ▷ C₃×C₁₁₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1172
transitive degree	μ_t(G) = 55344

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

Polycyclic presentation:

2^4,3,1153|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^1152

Conjugacy classes & character table

13839 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 1153]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^1152);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13836 #72290118