C₁₉×D₅₂₉ #34117313

order	20102
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×D₅₂₉
unambiguous semidirect product	C₅₂₉⋊(C₂×C₁₉)

Splittings

Fitting, Hall	C₁₉×C₅₂₉ ⋊ C₂
derived, Sylow	C₅₂₉ ⋊ C₂×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉	D₅₂₉
commutator, cosocle, hypocenter	C₅₂₉	C₂×C₁₉
Frattini	C₂₃	C₁₉×D₂₃
Fitting	C₁₉×C₅₂₉	C₂
socle	C₁₉×C₂₃	D₂₃
LCS	G ▷ C₅₂₉
UCS	C₁ ◁ C₁₉

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 548
transitive degree	μ_t(G) = 10051

A minimal faithful permutation representation — 2 generators on 548 points:

(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,20)(12,23)(14,25)(17,29)(18,28)(19,27)(21,34)(22,37)(24,39)(26,41)(30,46)(31,45)(32,44)(33,43)(35,52)(36,55)(38,57)(40,59)(42,61)(47,67)(48,66)(49,65)(50,64)(51,63)(53,74)(54,77)(56,79)(58,81)(60,83)(62,85)(68,92)(69,91)(70,90)(71,89)(72,88)(73,87)(75,100)(76,103)(78,105)(80,107)(82,109)(84,111)(86,113)(93,121)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(101,130)(102,133)(104,135)(106,137)(108,139)(110,141)(112,143)(114,145)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(131,164)(132,167)(134,169)(136,171)(138,173)(140,175)(142,177)(144,179)(146,181)(155,191)(156,190)(157,189)(158,188)(159,187)(160,186)(161,185)(162,184)(163,183)(165,202)(166,205)(168,207)(170,209)(172,211)(174,213)(176,215)(178,217)(180,219)(182,221)(192,232)(193,231)(194,230)(195,229)(196,228)(197,227)(198,226)(199,225)(200,224)(201,223)(203,244)(204,247)(206,249)(208,251)(210,253)(212,255)(214,257)(216,259)(218,261)(220,263)(222,265)(233,277)(234,276)(235,275)(236,274)(237,273)(238,272)(239,271)(240,270)(241,269)(242,268)(243,267)(245,288)(246,290)(248,292)(250,294)(252,296)(254,298)(256,300)(258,302)(260,304)(262,306)(264,308)(266,310)(278,311)(279,320)(280,319)(281,318)(282,317)(283,316)(284,315)(285,314)(286,313)(287,312)(289,330)(291,332)(293,334)(295,336)(297,338)(299,340)(301,342)(303,344)(305,346)(307,348)(309,350)(321,351)(322,359)(323,358)(324,357)(325,356)(326,355)(327,354)(328,353)(329,352)(331,368)(333,370)(335,372)(337,374)(339,376)(341,378)(343,380)(345,382)(347,384)(349,386)(360,387)(361,394)(362,393)(363,392)(364,391)(365,390)(366,389)(367,388)(369,402)(371,404)(373,406)(375,408)(377,410)(379,412)(381,414)(383,416)(385,418)(395,419)(396,425)(397,424)(398,423)(399,422)(400,421)(401,420)(403,432)(405,434)(407,436)(409,438)(411,440)(413,442)(415,444)(417,446)(426,447)(427,452)(428,451)(429,450)(430,449)(431,448)(433,458)(435,460)(437,462)(439,464)(441,466)(443,468)(445,470)(453,471)(454,475)(455,474)(456,473)(457,472)(459,480)(461,482)(463,484)(465,486)(467,488)(469,490)(476,491)(477,494)(478,493)(479,492)(481,498)(483,500)(485,502)(487,504)(489,506)(495,507)(496,509)(497,508)(499,512)(501,514)(503,516)(505,518)(510,519)(511,520)(513,522)(515,524)(517,526)(521,527)(523,528)(525,529)(530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548);(1,6,24,58,108,174,256,341,411,465,503,525,524,502,464,410,340,255,173,107,57,23,4,3,14,40,82,140,214,301,379,441,487,517,528,514,484,438,376,298,211,137,79,37,10,9,8,26,60,110,176,258,343,413,467,505,527,522,500,462,408,338,253,171,105,55,20,19,18,17,42,84,142,216,303,381,443,489,519,520,512,482,436,374,296,209,135,77,34,33,32,31,30,62,112,178,260,345,415,469,507,509,508,498,460,406,336,251,169,103,52,51,50,49,48,47,86,144,218,305,383,445,491,494,493,492,480,434,372,294,207,133,74,73,72,71,70,69,68,114,180,262,347,417,471,475,474,473,472,458,404,334,249,167,100,99,98,97,96,95,94,93,146,220,307,385,447,452,451,450,449,448,432,370,292,205,130,129,128,127,126,125,124,123,122,182,264,349,419,425,424,423,422,421,420,402,332,247,164,163,162,161,160,159,158,157,156,155,222,309,387,394,393,392,391,390,389,388,368,290,202,201,200,199,198,197,196,195,194,193,192,266,351,359,358,357,356,355,354,353,352,330,244,243,242,241,240,239,238,237,236,235,234,233,311,320,319,318,317,316,315,314,313,312,288,245,287,286,285,284,283,282,281,280,279,278,277,276,275,274,273,272,271,270,269,268,267,203,289,329,328,327,326,325,324,323,322,321,310,232,231,230,229,228,227,226,225,224,223,165,246,331,367,366,365,364,363,362,361,360,350,265,191,190,189,188,187,186,185,184,183,131,204,291,369,401,400,399,398,397,396,395,386,308,221,154,153,152,151,150,149,148,147,101,166,248,333,403,431,430,429,428,427,426,418,348,263,181,121,120,119,118,117,116,115,75,132,206,293,371,433,457,456,455,454,453,446,384,306,219,145,92,91,90,89,88,87,53,102,168,250,335,405,459,479,478,477,476,470,416,346,261,179,113,67,66,65,64,63,35,76,134,208,295,373,435,481,497,496,495,490,444,382,304,217,143,85,46,45,44,43,21,54,104,170,252,337,407,461,499,511,510,506,468,414,344,259,177,111,61,29,28,27,11,36,78,136,210,297,375,437,483,513,521,518,488,442,380,302,215,141,83,41,16,15,5,22,56,106,172,254,339,409,463,501,523,526,504,466,412,342,257,175,109,59,25,7,2,12,38,80,138,212,299,377,439,485,515,529,516,486,440,378,300,213,139,81,39,13)

Conjugacy classes & character table

5054 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20102, 3);

C19×D529 #34117313