Finite Group DatabasebetaFinite Group Databasebeta| order | 20052 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C3⋊#14555805) |
| Fitting, socle, derived, Hall | C32×C557 ⋊ C4 |
| Sylow | C557 ⋊ C3⋊Dic3 |
| Sylow | C32 ⋊ #14555805 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C557 | C4 |
| prime radical, cosocle | C32×D557 | C2 |
| LCS | G ▷ C32×C557 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 563 |
| transitive degree | μt(G) = 5013 |
A minimal faithful permutation representation — 3 generators on 563 points:
(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,91)(53,72,73,87)(57,76,77,83)(61,80,81,79)(65,84,85,75)(67,88,89,71)(70,93,94,120)(74,97,98,116)(78,101,102,112)(82,105,106,108)(86,109,110,104)(90,113,114,100)(92,117,118,96)(95,122,123,153)(99,126,127,149)(103,130,131,145)(107,134,135,141)(111,138,139,137)(115,142,143,133)(119,146,147,129)(121,150,151,125)(124,155,156,190)(128,159,160,186)(132,163,164,182)(136,167,168,178)(140,171,172,174)(144,175,176,170)(148,179,180,166)(152,183,184,162)(154,187,188,158)(157,192,193,231)(161,196,197,227)(165,200,201,223)(169,204,205,219)(173,208,209,215)(177,212,213,211)(181,216,217,207)(185,220,221,203)(189,224,225,199)(191,228,229,195)(194,233,234,276)(198,237,238,272)(202,241,242,268)(206,245,246,264)(210,249,250,260)(214,253,254,256)(218,257,258,252)(222,261,262,248)(226,265,266,244)(230,269,270,240)(232,273,274,236)(235,278,279,325)(239,282,283,321)(243,286,287,317)(247,290,291,313)(251,294,295,309)(255,298,299,305)(259,302,303,301)(263,306,307,297)(267,310,311,293)(271,314,315,289)(275,318,319,285)(277,322,323,281)(280,327,328,378)(284,331,332,374)(288,335,336,370)(292,339,340,366)(296,343,344,362)(300,347,348,358)(304,351,352,354)(308,355,356,350)(312,359,360,346)(316,363,364,342)(320,367,368,338)(324,371,372,334)(326,375,376,330)(329,380,381,435)(333,384,385,431)(337,388,389,427)(341,392,393,423)(345,396,397,419)(349,400,401,415)(353,404,405,411)(357,408,409,407)(361,412,413,403)(365,416,417,399)(369,420,421,395)(373,424,425,391)(377,428,429,387)(379,432,433,383)(382,437,438,496)(386,441,442,492)(390,445,446,488)(394,449,450,484)(398,453,454,480)(402,457,458,476)(406,461,462,472)(410,465,466,468)(414,469,470,464)(418,473,474,460)(422,477,478,456)(426,481,482,452)(430,485,486,448)(434,489,490,444)(436,493,494,440)(439,498,499,511)(443,502,503,546)(447,500,506,544)(451,509,510,542)(455,513,514,540)(459,517,518,538)(463,521,522,536)(467,525,526,532)(471,529,530,528)(475,533,534,524)(479,537,535,520)(483,539,531,516)(487,541,527,512)(491,543,523,508)(495,545,519,505)(497,547,515,501)(504,548,549,553)(507,550,551,554)(552,555,556,557)(559,560)(562,563);(1,2,5,10,19,32,49,70,95,124,157,194,235,280,329,382,439,500,441,380,436,495,518,458,401,348,299,254,213,176,143,114,89,66,87,112,141,174,211,252,297,346,399,456,516,525,461,400,343,290,241,196,155,191,230,271,316,365,418,475,535,478,421,368,319,274,231,272,317,366,419,476,536,541,481,420,363,310,261,216,175,138,105,76,51,30,46,65,86,111,140,173,210,251,296,345,398,455,515,496,546,554,549,503,442,385,332,283,238,197,160,127,98,73,52,35,22,13,6,11,20,33,50,71,96,125,158,195,236,281,330,383,440,501,509,445,384,327,379,434,491,522,462,405,352,303,258,217,180,147,118,91,116,145,178,215,256,301,350,403,460,520,529,465,404,347,294,245,200,159,122,154,189,226,267,312,361,414,471,531,482,425,372,323,276,321,370,423,480,538,543,485,424,367,314,265,220,179,142,109,80,55,34,17,29,44,61,82,107,136,169,206,247,292,341,394,451,511,553,557,556,551,506,446,389,336,287,242,201,164,131,102,77,56,39,26,15,24,37,54,75,100,129,162,199,240,285,334,387,444,505,513,449,388,331,278,326,377,430,487,526,466,409,356,307,262,221,184,151,120,149,182,219,260,305,354,407,464,524,533,469,408,351,298,249,204,163,126,93,121,152,185,222,263,308,357,410,467,527,486,429,376,325,374,427,484,540,545,489,428,371,318,269,224,183,146,113,84,59,38,21,8,16,27,40,57,78,103,132,165,202,243,288,337,390,447,507,552,555,548,498,510,450,393,340,291,246,205,168,135,106,81,60,43,28,41,58,79,104,133,166,203,244,289,338,391,448,508,517,453,392,335,282,233,277,324,373,426,483,530,470,413,360,311,266,225,188,153,186,223,264,309,358,411,468,528,537,473,412,355,302,253,208,167,130,97,68,92,119,148,181,218,259,304,353,406,463,523,490,433,378,431,488,542,547,493,432,375,322,273,228,187,150,117,88,63,42,25,12,3,7,14,23,36,53,74,99,128,161,198,239,284,333,386,443,504,550,502,437,497,514,454,397,344,295,250,209,172,139,110,85,64,45,62,83,108,137,170,207,248,293,342,395,452,512,521,457,396,339,286,237,192,232,275,320,369,422,479,534,474,417,364,315,270,229,190,227,268,313,362,415,472,532,539,477,416,359,306,257,212,171,134,101,72,47,67,90,115,144,177,214,255,300,349,402,459,519,494,435,492,544,499,438,381,328,279,234,193,156,123,94,69,48,31,18,9,4)(561,562,563);(558,559,560)Polycyclic presentation:
2^2,3^2,557|a^2=b,c^a=c^2,d^a=d^2,e^a=e^439,e^b=e^5561263 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20052, 29);