D₁₁₉₈₉ #37561824

order	23978
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₉₈₉
unambiguous semidirect product	C₁₉⋊D₆₃₁
unambiguous semidirect product	C₆₃₁⋊D₁₉

Splittings

Fitting, socle, derived, Hall	C₁₉×C₆₃₁ ⋊ C₂
Sylow	C₆₃₁ ⋊ D₁₉
Sylow	C₁₉ ⋊ D₆₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₉×C₆₃₁	C₂
LCS	G ▷ C₁₉×C₆₃₁
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 650
transitive degree	μ_t(G) = 11989

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

Polycyclic presentation:

2,19,631|b^a=b^18,c^a=c^630

Conjugacy classes & character table

5996 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23978, 3);

D11989 #37561824