D₁₁₅₃₃ #37528022

order	23066
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₅₃₃
unambiguous semidirect product	C₁₉⋊D₆₀₇
unambiguous semidirect product	C₆₀₇⋊D₁₉

Splittings

Fitting, socle, derived, Hall	C₁₉×C₆₀₇ ⋊ C₂
Sylow	C₆₀₇ ⋊ D₁₉
Sylow	C₁₉ ⋊ D₆₀₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₉×C₆₀₇	C₂
LCS	G ▷ C₁₉×C₆₀₇
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 626
transitive degree	μ_t(G) = 11533

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

Polycyclic presentation:

2,19,607|b^a=b^18,c^a=c^606

Conjugacy classes & character table

5768 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23066, 3);

D11533 #37528022