D₁₇₇₉₃ #72273293

order	35586
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₁₇₇₉₃

Splittings

Fitting, derived, Hall	C₂₇×C₆₅₉ ⋊ C₂
Sylow	C₆₅₉ ⋊ D₂₇
Sylow	C₂₇ ⋊ D₆₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₇×C₆₅₉	C₂
Frattini	C₉	D₁₉₇₇
socle	C₃×C₆₅₉	D₉
LCS	G ▷ C₂₇×C₆₅₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 686
transitive degree	μ_t(G) = 17793

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

Polycyclic presentation:

2,3,659,3^2|b^3=d,d^3=e,b^a=b^2d^2e^2,c^a=c^658,d^a=d^2e^2,e^a=e^2

Conjugacy classes & character table

8898 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, 659, 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.3^658);;
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);

D17793 #72273293