D₂₇₂₉₃ #72290613

order	54586
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₇₂₉₃

Splittings

Fitting, derived, Hall	C₄₉×C₅₅₇ ⋊ C₂
Sylow	C₅₅₇ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₅₅₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄₉×C₅₅₇	C₂
Frattini	C₇	D₃₈₉₉
socle	C₇×C₅₅₇	D₇
LCS	G ▷ C₄₉×C₅₅₇
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 606
transitive degree	μ_t(G) = 27293

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

Polycyclic presentation:

2,7,557,7|b^7=d,b^a=b^6d^6,c^a=c^556,d^a=d^6

Conjugacy classes & character table

13648 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, 557, 7]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2^6*F.4^6);;
gap> SetConjugate(coll, 3, 1, F.3^556);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27293 #72290613