Finite Group Databasebeta

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D₁₈₂₇₉ #72273496

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

Aliases

Splittings

Subgroups

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 704 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);(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,704)(30,703)(31,702)(32,701)(33,700)(34,699)(35,698)(36,697)(37,696)(38,695)(39,694)(40,693)(41,692)(42,691)(43,690)(44,689)(45,688)(46,687)(47,686)(48,685)(49,684)(50,683)(51,682)(52,681)(53,680)(54,679)(55,678)(56,677)(57,676)(58,675)(59,674)(60,673)(61,672)(62,671)(63,670)(64,669)(65,668)(66,667)(67,666)(68,665)(69,664)(70,663)(71,662)(72,661)(73,660)(74,659)(75,658)(76,657)(77,656)(78,655)(79,654)(80,653)(81,652)(82,651)(83,650)(84,649)(85,648)(86,647)(87,646)(88,645)(89,644)(90,643)(91,642)(92,641)(93,640)(94,639)(95,638)(96,637)(97,636)(98,635)(99,634)(100,633)(101,632)(102,631)(103,630)(104,629)(105,628)(106,627)(107,626)(108,625)(109,624)(110,623)(111,622)(112,621)(113,620)(114,619)(115,618)(116,617)(117,616)(118,615)(119,614)(120,613)(121,612)(122,611)(123,610)(124,609)(125,608)(126,607)(127,606)(128,605)(129,604)(130,603)(131,602)(132,601)(133,600)(134,599)(135,598)(136,597)(137,596)(138,595)(139,594)(140,593)(141,592)(142,591)(143,590)(144,589)(145,588)(146,587)(147,586)(148,585)(149,584)(150,583)(151,582)(152,581)(153,580)(154,579)(155,578)(156,577)(157,576)(158,575)(159,574)(160,573)(161,572)(162,571)(163,570)(164,569)(165,568)(166,567)(167,566)(168,565)(169,564)(170,563)(171,562)(172,561)(173,560)(174,559)(175,558)(176,557)(177,556)(178,555)(179,554)(180,553)(181,552)(182,551)(183,550)(184,549)(185,548)(186,547)(187,546)(188,545)(189,544)(190,543)(191,542)(192,541)(193,540)(194,539)(195,538)(196,537)(197,536)(198,535)(199,534)(200,533)(201,532)(202,531)(203,530)(204,529)(205,528)(206,527)(207,526)(208,525)(209,524)(210,523)(211,522)(212,521)(213,520)(214,519)(215,518)(216,517)(217,516)(218,515)(219,514)(220,513)(221,512)(222,511)(223,510)(224,509)(225,508)(226,507)(227,506)(228,505)(229,504)(230,503)(231,502)(232,501)(233,500)(234,499)(235,498)(236,497)(237,496)(238,495)(239,494)(240,493)(241,492)(242,491)(243,490)(244,489)(245,488)(246,487)(247,486)(248,485)(249,484)(250,483)(251,482)(252,481)(253,480)(254,479)(255,478)(256,477)(257,476)(258,475)(259,474)(260,473)(261,472)(262,471)(263,470)(264,469)(265,468)(266,467)(267,466)(268,465)(269,464)(270,463)(271,462)(272,461)(273,460)(274,459)(275,458)(276,457)(277,456)(278,455)(279,454)(280,453)(281,452)(282,451)(283,450)(284,449)(285,448)(286,447)(287,446)(288,445)(289,444)(290,443)(291,442)(292,441)(293,440)(294,439)(295,438)(296,437)(297,436)(298,435)(299,434)(300,433)(301,432)(302,431)(303,430)(304,429)(305,428)(306,427)(307,426)(308,425)(309,424)(310,423)(311,422)(312,421)(313,420)(314,419)(315,418)(316,417)(317,416)(318,415)(319,414)(320,413)(321,412)(322,411)(323,410)(324,409)(325,408)(326,407)(327,406)(328,405)(329,404)(330,403)(331,402)(332,401)(333,400)(334,399)(335,398)(336,397)(337,396)(338,395)(339,394)(340,393)(341,392)(342,391)(343,390)(344,389)(345,388)(346,387)(347,386)(348,385)(349,384)(350,383)(351,382)(352,381)(353,380)(354,379)(355,378)(356,377)(357,376)(358,375)(359,374)(360,373)(361,372)(362,371)(363,370)(364,369)(365,368)(366,367)

Polycyclic presentation:

2,3,677,3^2|b^3=d,d^3=e,b^a=b^2d^2e^2,c^a=c^676,d^a=d^2e^2,e^a=e^2

Conjugacy classes & character table

9141 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, 3, 677, 3, 3]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.4^2*F.5^2);;
gap> SetConjugate(coll, 3, 1, F.3^676);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.5^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);