Dic₁₅₉₄₁ #72296745

order	63764
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₉₄₁

Splittings

derived, Hall	C₁₉×C₈₃₉ ⋊ C₄
Sylow	C₈₃₉ ⋊ Dic₁₉
Sylow	C₁₉ ⋊ Dic₈₃₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅₉₄₁
commutator, hypocenter	C₁₉×C₈₃₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₉×C₈₃₉	C₂
LCS	G ▷ C₁₉×C₈₃₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 862
transitive degree	μ_t(G) = 63764

A minimal faithful permutation representation — 2 generators on 862 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);(1,3,2,4)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(25,862)(26,861)(27,860)(28,859)(29,858)(30,857)(31,856)(32,855)(33,854)(34,853)(35,852)(36,851)(37,850)(38,849)(39,848)(40,847)(41,846)(42,845)(43,844)(44,843)(45,842)(46,841)(47,840)(48,839)(49,838)(50,837)(51,836)(52,835)(53,834)(54,833)(55,832)(56,831)(57,830)(58,829)(59,828)(60,827)(61,826)(62,825)(63,824)(64,823)(65,822)(66,821)(67,820)(68,819)(69,818)(70,817)(71,816)(72,815)(73,814)(74,813)(75,812)(76,811)(77,810)(78,809)(79,808)(80,807)(81,806)(82,805)(83,804)(84,803)(85,802)(86,801)(87,800)(88,799)(89,798)(90,797)(91,796)(92,795)(93,794)(94,793)(95,792)(96,791)(97,790)(98,789)(99,788)(100,787)(101,786)(102,785)(103,784)(104,783)(105,782)(106,781)(107,780)(108,779)(109,778)(110,777)(111,776)(112,775)(113,774)(114,773)(115,772)(116,771)(117,770)(118,769)(119,768)(120,767)(121,766)(122,765)(123,764)(124,763)(125,762)(126,761)(127,760)(128,759)(129,758)(130,757)(131,756)(132,755)(133,754)(134,753)(135,752)(136,751)(137,750)(138,749)(139,748)(140,747)(141,746)(142,745)(143,744)(144,743)(145,742)(146,741)(147,740)(148,739)(149,738)(150,737)(151,736)(152,735)(153,734)(154,733)(155,732)(156,731)(157,730)(158,729)(159,728)(160,727)(161,726)(162,725)(163,724)(164,723)(165,722)(166,721)(167,720)(168,719)(169,718)(170,717)(171,716)(172,715)(173,714)(174,713)(175,712)(176,711)(177,710)(178,709)(179,708)(180,707)(181,706)(182,705)(183,704)(184,703)(185,702)(186,701)(187,700)(188,699)(189,698)(190,697)(191,696)(192,695)(193,694)(194,693)(195,692)(196,691)(197,690)(198,689)(199,688)(200,687)(201,686)(202,685)(203,684)(204,683)(205,682)(206,681)(207,680)(208,679)(209,678)(210,677)(211,676)(212,675)(213,674)(214,673)(215,672)(216,671)(217,670)(218,669)(219,668)(220,667)(221,666)(222,665)(223,664)(224,663)(225,662)(226,661)(227,660)(228,659)(229,658)(230,657)(231,656)(232,655)(233,654)(234,653)(235,652)(236,651)(237,650)(238,649)(239,648)(240,647)(241,646)(242,645)(243,644)(244,643)(245,642)(246,641)(247,640)(248,639)(249,638)(250,637)(251,636)(252,635)(253,634)(254,633)(255,632)(256,631)(257,630)(258,629)(259,628)(260,627)(261,626)(262,625)(263,624)(264,623)(265,622)(266,621)(267,620)(268,619)(269,618)(270,617)(271,616)(272,615)(273,614)(274,613)(275,612)(276,611)(277,610)(278,609)(279,608)(280,607)(281,606)(282,605)(283,604)(284,603)(285,602)(286,601)(287,600)(288,599)(289,598)(290,597)(291,596)(292,595)(293,594)(294,593)(295,592)(296,591)(297,590)(298,589)(299,588)(300,587)(301,586)(302,585)(303,584)(304,583)(305,582)(306,581)(307,580)(308,579)(309,578)(310,577)(311,576)(312,575)(313,574)(314,573)(315,572)(316,571)(317,570)(318,569)(319,568)(320,567)(321,566)(322,565)(323,564)(324,563)(325,562)(326,561)(327,560)(328,559)(329,558)(330,557)(331,556)(332,555)(333,554)(334,553)(335,552)(336,551)(337,550)(338,549)(339,548)(340,547)(341,546)(342,545)(343,544)(344,543)(345,542)(346,541)(347,540)(348,539)(349,538)(350,537)(351,536)(352,535)(353,534)(354,533)(355,532)(356,531)(357,530)(358,529)(359,528)(360,527)(361,526)(362,525)(363,524)(364,523)(365,522)(366,521)(367,520)(368,519)(369,518)(370,517)(371,516)(372,515)(373,514)(374,513)(375,512)(376,511)(377,510)(378,509)(379,508)(380,507)(381,506)(382,505)(383,504)(384,503)(385,502)(386,501)(387,500)(388,499)(389,498)(390,497)(391,496)(392,495)(393,494)(394,493)(395,492)(396,491)(397,490)(398,489)(399,488)(400,487)(401,486)(402,485)(403,484)(404,483)(405,482)(406,481)(407,480)(408,479)(409,478)(410,477)(411,476)(412,475)(413,474)(414,473)(415,472)(416,471)(417,470)(418,469)(419,468)(420,467)(421,466)(422,465)(423,464)(424,463)(425,462)(426,461)(427,460)(428,459)(429,458)(430,457)(431,456)(432,455)(433,454)(434,453)(435,452)(436,451)(437,450)(438,449)(439,448)(440,447)(441,446)(442,445)(443,444)

Polycyclic presentation:

2^2,19,839|a^2=b,c^a=c^18,d^a=d^838

Conjugacy classes & character table

15944 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");;
gap> coll := SingleCollector(F, [2, 2, 19, 839]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^18);;
gap> SetConjugate(coll, 4, 1, F.4^838);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15941 #72296745