D₁₂₂₉₃ #37589639

order	24586
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₂₉₃
unambiguous semidirect product	C₁₉⋊D₆₄₇
unambiguous semidirect product	C₆₄₇⋊D₁₉

Splittings

Fitting, socle, derived, Hall	C₁₉×C₆₄₇ ⋊ C₂
Sylow	C₆₄₇ ⋊ D₁₉
Sylow	C₁₉ ⋊ D₆₄₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₉×C₆₄₇	C₂
LCS	G ▷ C₁₉×C₆₄₇
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 666
transitive degree	μ_t(G) = 12293

A minimal faithful permutation representation — 2 generators on 666 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);(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,666)(22,665)(23,664)(24,663)(25,662)(26,661)(27,660)(28,659)(29,658)(30,657)(31,656)(32,655)(33,654)(34,653)(35,652)(36,651)(37,650)(38,649)(39,648)(40,647)(41,646)(42,645)(43,644)(44,643)(45,642)(46,641)(47,640)(48,639)(49,638)(50,637)(51,636)(52,635)(53,634)(54,633)(55,632)(56,631)(57,630)(58,629)(59,628)(60,627)(61,626)(62,625)(63,624)(64,623)(65,622)(66,621)(67,620)(68,619)(69,618)(70,617)(71,616)(72,615)(73,614)(74,613)(75,612)(76,611)(77,610)(78,609)(79,608)(80,607)(81,606)(82,605)(83,604)(84,603)(85,602)(86,601)(87,600)(88,599)(89,598)(90,597)(91,596)(92,595)(93,594)(94,593)(95,592)(96,591)(97,590)(98,589)(99,588)(100,587)(101,586)(102,585)(103,584)(104,583)(105,582)(106,581)(107,580)(108,579)(109,578)(110,577)(111,576)(112,575)(113,574)(114,573)(115,572)(116,571)(117,570)(118,569)(119,568)(120,567)(121,566)(122,565)(123,564)(124,563)(125,562)(126,561)(127,560)(128,559)(129,558)(130,557)(131,556)(132,555)(133,554)(134,553)(135,552)(136,551)(137,550)(138,549)(139,548)(140,547)(141,546)(142,545)(143,544)(144,543)(145,542)(146,541)(147,540)(148,539)(149,538)(150,537)(151,536)(152,535)(153,534)(154,533)(155,532)(156,531)(157,530)(158,529)(159,528)(160,527)(161,526)(162,525)(163,524)(164,523)(165,522)(166,521)(167,520)(168,519)(169,518)(170,517)(171,516)(172,515)(173,514)(174,513)(175,512)(176,511)(177,510)(178,509)(179,508)(180,507)(181,506)(182,505)(183,504)(184,503)(185,502)(186,501)(187,500)(188,499)(189,498)(190,497)(191,496)(192,495)(193,494)(194,493)(195,492)(196,491)(197,490)(198,489)(199,488)(200,487)(201,486)(202,485)(203,484)(204,483)(205,482)(206,481)(207,480)(208,479)(209,478)(210,477)(211,476)(212,475)(213,474)(214,473)(215,472)(216,471)(217,470)(218,469)(219,468)(220,467)(221,466)(222,465)(223,464)(224,463)(225,462)(226,461)(227,460)(228,459)(229,458)(230,457)(231,456)(232,455)(233,454)(234,453)(235,452)(236,451)(237,450)(238,449)(239,448)(240,447)(241,446)(242,445)(243,444)(244,443)(245,442)(246,441)(247,440)(248,439)(249,438)(250,437)(251,436)(252,435)(253,434)(254,433)(255,432)(256,431)(257,430)(258,429)(259,428)(260,427)(261,426)(262,425)(263,424)(264,423)(265,422)(266,421)(267,420)(268,419)(269,418)(270,417)(271,416)(272,415)(273,414)(274,413)(275,412)(276,411)(277,410)(278,409)(279,408)(280,407)(281,406)(282,405)(283,404)(284,403)(285,402)(286,401)(287,400)(288,399)(289,398)(290,397)(291,396)(292,395)(293,394)(294,393)(295,392)(296,391)(297,390)(298,389)(299,388)(300,387)(301,386)(302,385)(303,384)(304,383)(305,382)(306,381)(307,380)(308,379)(309,378)(310,377)(311,376)(312,375)(313,374)(314,373)(315,372)(316,371)(317,370)(318,369)(319,368)(320,367)(321,366)(322,365)(323,364)(324,363)(325,362)(326,361)(327,360)(328,359)(329,358)(330,357)(331,356)(332,355)(333,354)(334,353)(335,352)(336,351)(337,350)(338,349)(339,348)(340,347)(341,346)(342,345)(343,344)

Polycyclic presentation:

2,19,647|b^a=b^18,c^a=c^646

Conjugacy classes & character table

6148 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24586, 5);

D12293 #37589639