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D₂₃₂₁₂ #72279745

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

Aliases

Splittings

Subgroups

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 840 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);(2,4)(6,11)(7,10)(8,9)(13,840)(14,839)(15,838)(16,837)(17,836)(18,835)(19,834)(20,833)(21,832)(22,831)(23,830)(24,829)(25,828)(26,827)(27,826)(28,825)(29,824)(30,823)(31,822)(32,821)(33,820)(34,819)(35,818)(36,817)(37,816)(38,815)(39,814)(40,813)(41,812)(42,811)(43,810)(44,809)(45,808)(46,807)(47,806)(48,805)(49,804)(50,803)(51,802)(52,801)(53,800)(54,799)(55,798)(56,797)(57,796)(58,795)(59,794)(60,793)(61,792)(62,791)(63,790)(64,789)(65,788)(66,787)(67,786)(68,785)(69,784)(70,783)(71,782)(72,781)(73,780)(74,779)(75,778)(76,777)(77,776)(78,775)(79,774)(80,773)(81,772)(82,771)(83,770)(84,769)(85,768)(86,767)(87,766)(88,765)(89,764)(90,763)(91,762)(92,761)(93,760)(94,759)(95,758)(96,757)(97,756)(98,755)(99,754)(100,753)(101,752)(102,751)(103,750)(104,749)(105,748)(106,747)(107,746)(108,745)(109,744)(110,743)(111,742)(112,741)(113,740)(114,739)(115,738)(116,737)(117,736)(118,735)(119,734)(120,733)(121,732)(122,731)(123,730)(124,729)(125,728)(126,727)(127,726)(128,725)(129,724)(130,723)(131,722)(132,721)(133,720)(134,719)(135,718)(136,717)(137,716)(138,715)(139,714)(140,713)(141,712)(142,711)(143,710)(144,709)(145,708)(146,707)(147,706)(148,705)(149,704)(150,703)(151,702)(152,701)(153,700)(154,699)(155,698)(156,697)(157,696)(158,695)(159,694)(160,693)(161,692)(162,691)(163,690)(164,689)(165,688)(166,687)(167,686)(168,685)(169,684)(170,683)(171,682)(172,681)(173,680)(174,679)(175,678)(176,677)(177,676)(178,675)(179,674)(180,673)(181,672)(182,671)(183,670)(184,669)(185,668)(186,667)(187,666)(188,665)(189,664)(190,663)(191,662)(192,661)(193,660)(194,659)(195,658)(196,657)(197,656)(198,655)(199,654)(200,653)(201,652)(202,651)(203,650)(204,649)(205,648)(206,647)(207,646)(208,645)(209,644)(210,643)(211,642)(212,641)(213,640)(214,639)(215,638)(216,637)(217,636)(218,635)(219,634)(220,633)(221,632)(222,631)(223,630)(224,629)(225,628)(226,627)(227,626)(228,625)(229,624)(230,623)(231,622)(232,621)(233,620)(234,619)(235,618)(236,617)(237,616)(238,615)(239,614)(240,613)(241,612)(242,611)(243,610)(244,609)(245,608)(246,607)(247,606)(248,605)(249,604)(250,603)(251,602)(252,601)(253,600)(254,599)(255,598)(256,597)(257,596)(258,595)(259,594)(260,593)(261,592)(262,591)(263,590)(264,589)(265,588)(266,587)(267,586)(268,585)(269,584)(270,583)(271,582)(272,581)(273,580)(274,579)(275,578)(276,577)(277,576)(278,575)(279,574)(280,573)(281,572)(282,571)(283,570)(284,569)(285,568)(286,567)(287,566)(288,565)(289,564)(290,563)(291,562)(292,561)(293,560)(294,559)(295,558)(296,557)(297,556)(298,555)(299,554)(300,553)(301,552)(302,551)(303,550)(304,549)(305,548)(306,547)(307,546)(308,545)(309,544)(310,543)(311,542)(312,541)(313,540)(314,539)(315,538)(316,537)(317,536)(318,535)(319,534)(320,533)(321,532)(322,531)(323,530)(324,529)(325,528)(326,527)(327,526)(328,525)(329,524)(330,523)(331,522)(332,521)(333,520)(334,519)(335,518)(336,517)(337,516)(338,515)(339,514)(340,513)(341,512)(342,511)(343,510)(344,509)(345,508)(346,507)(347,506)(348,505)(349,504)(350,503)(351,502)(352,501)(353,500)(354,499)(355,498)(356,497)(357,496)(358,495)(359,494)(360,493)(361,492)(362,491)(363,490)(364,489)(365,488)(366,487)(367,486)(368,485)(369,484)(370,483)(371,482)(372,481)(373,480)(374,479)(375,478)(376,477)(377,476)(378,475)(379,474)(380,473)(381,472)(382,471)(383,470)(384,469)(385,468)(386,467)(387,466)(388,465)(389,464)(390,463)(391,462)(392,461)(393,460)(394,459)(395,458)(396,457)(397,456)(398,455)(399,454)(400,453)(401,452)(402,451)(403,450)(404,449)(405,448)(406,447)(407,446)(408,445)(409,444)(410,443)(411,442)(412,441)(413,440)(414,439)(415,438)(416,437)(417,436)(418,435)(419,434)(420,433)(421,432)(422,431)(423,430)(424,429)(425,428)(426,427)

Polycyclic presentation:

2^3,7,829|b^2=c,b^a=bc,d^a=d^6,e^a=e^828

Conjugacy classes & character table

11609 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 7, 829]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^828);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);