D₂₉₇₄₃ #72293704

order	59486
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₉₇₄₃

Splittings

Fitting, derived, Hall	C₄₉×C₆₀₇ ⋊ C₂
Sylow	C₆₀₇ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₆₀₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄₉×C₆₀₇	C₂
Frattini	C₇	D₄₂₄₉
socle	C₇×C₆₀₇	D₇
LCS	G ▷ C₄₉×C₆₀₇
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 656
transitive degree	μ_t(G) = 29743

A minimal faithful permutation representation — 2 generators on 656 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);(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(51,656)(52,655)(53,654)(54,653)(55,652)(56,651)(57,650)(58,649)(59,648)(60,647)(61,646)(62,645)(63,644)(64,643)(65,642)(66,641)(67,640)(68,639)(69,638)(70,637)(71,636)(72,635)(73,634)(74,633)(75,632)(76,631)(77,630)(78,629)(79,628)(80,627)(81,626)(82,625)(83,624)(84,623)(85,622)(86,621)(87,620)(88,619)(89,618)(90,617)(91,616)(92,615)(93,614)(94,613)(95,612)(96,611)(97,610)(98,609)(99,608)(100,607)(101,606)(102,605)(103,604)(104,603)(105,602)(106,601)(107,600)(108,599)(109,598)(110,597)(111,596)(112,595)(113,594)(114,593)(115,592)(116,591)(117,590)(118,589)(119,588)(120,587)(121,586)(122,585)(123,584)(124,583)(125,582)(126,581)(127,580)(128,579)(129,578)(130,577)(131,576)(132,575)(133,574)(134,573)(135,572)(136,571)(137,570)(138,569)(139,568)(140,567)(141,566)(142,565)(143,564)(144,563)(145,562)(146,561)(147,560)(148,559)(149,558)(150,557)(151,556)(152,555)(153,554)(154,553)(155,552)(156,551)(157,550)(158,549)(159,548)(160,547)(161,546)(162,545)(163,544)(164,543)(165,542)(166,541)(167,540)(168,539)(169,538)(170,537)(171,536)(172,535)(173,534)(174,533)(175,532)(176,531)(177,530)(178,529)(179,528)(180,527)(181,526)(182,525)(183,524)(184,523)(185,522)(186,521)(187,520)(188,519)(189,518)(190,517)(191,516)(192,515)(193,514)(194,513)(195,512)(196,511)(197,510)(198,509)(199,508)(200,507)(201,506)(202,505)(203,504)(204,503)(205,502)(206,501)(207,500)(208,499)(209,498)(210,497)(211,496)(212,495)(213,494)(214,493)(215,492)(216,491)(217,490)(218,489)(219,488)(220,487)(221,486)(222,485)(223,484)(224,483)(225,482)(226,481)(227,480)(228,479)(229,478)(230,477)(231,476)(232,475)(233,474)(234,473)(235,472)(236,471)(237,470)(238,469)(239,468)(240,467)(241,466)(242,465)(243,464)(244,463)(245,462)(246,461)(247,460)(248,459)(249,458)(250,457)(251,456)(252,455)(253,454)(254,453)(255,452)(256,451)(257,450)(258,449)(259,448)(260,447)(261,446)(262,445)(263,444)(264,443)(265,442)(266,441)(267,440)(268,439)(269,438)(270,437)(271,436)(272,435)(273,434)(274,433)(275,432)(276,431)(277,430)(278,429)(279,428)(280,427)(281,426)(282,425)(283,424)(284,423)(285,422)(286,421)(287,420)(288,419)(289,418)(290,417)(291,416)(292,415)(293,414)(294,413)(295,412)(296,411)(297,410)(298,409)(299,408)(300,407)(301,406)(302,405)(303,404)(304,403)(305,402)(306,401)(307,400)(308,399)(309,398)(310,397)(311,396)(312,395)(313,394)(314,393)(315,392)(316,391)(317,390)(318,389)(319,388)(320,387)(321,386)(322,385)(323,384)(324,383)(325,382)(326,381)(327,380)(328,379)(329,378)(330,377)(331,376)(332,375)(333,374)(334,373)(335,372)(336,371)(337,370)(338,369)(339,368)(340,367)(341,366)(342,365)(343,364)(344,363)(345,362)(346,361)(347,360)(348,359)(349,358)(350,357)(351,356)(352,355)(353,354)

Polycyclic presentation:

2,7,607,7|b^7=d,b^a=b^6d^6,c^a=c^606,d^a=d^6

Conjugacy classes & character table

14873 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, 7, 607, 7]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2^6*F.4^6);;
gap> SetConjugate(coll, 3, 1, F.3^606);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D29743 #72293704