C₃×D₅₅₇₉ #45691049

order	33474
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₅₅₇₉
unambiguous semidirect product	C₇₉₇⋊(C₃×D₇)

Splittings

Fitting, socle, Hall	C₃×C₇×C₇₉₇ ⋊ C₂
derived, Hall	C₇×C₇₉₇ ⋊ C₂×C₃
Hall	C₃×C₇₉₇ ⋊ D₇
Sylow	C₇₉₇ ⋊ C₃×D₇
Hall	C₃×C₇ ⋊ D₇₉₇
Sylow	C₇ ⋊ C₃×D₇₉₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₅₅₇₉
commutator, cosocle, hypocenter	C₇×C₇₉₇	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₇₉₇	C₂
LCS	G ▷ C₇×C₇₉₇
UCS	C₁ ◁ C₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 807
transitive degree	μ_t(G) = 16737

A minimal faithful permutation representation — 2 generators on 807 points:

(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807);(5,10)(6,9)(7,8)(12,807)(13,806)(14,805)(15,804)(16,803)(17,802)(18,801)(19,800)(20,799)(21,798)(22,797)(23,796)(24,795)(25,794)(26,793)(27,792)(28,791)(29,790)(30,789)(31,788)(32,787)(33,786)(34,785)(35,784)(36,783)(37,782)(38,781)(39,780)(40,779)(41,778)(42,777)(43,776)(44,775)(45,774)(46,773)(47,772)(48,771)(49,770)(50,769)(51,768)(52,767)(53,766)(54,765)(55,764)(56,763)(57,762)(58,761)(59,760)(60,759)(61,758)(62,757)(63,756)(64,755)(65,754)(66,753)(67,752)(68,751)(69,750)(70,749)(71,748)(72,747)(73,746)(74,745)(75,744)(76,743)(77,742)(78,741)(79,740)(80,739)(81,738)(82,737)(83,736)(84,735)(85,734)(86,733)(87,732)(88,731)(89,730)(90,729)(91,728)(92,727)(93,726)(94,725)(95,724)(96,723)(97,722)(98,721)(99,720)(100,719)(101,718)(102,717)(103,716)(104,715)(105,714)(106,713)(107,712)(108,711)(109,710)(110,709)(111,708)(112,707)(113,706)(114,705)(115,704)(116,703)(117,702)(118,701)(119,700)(120,699)(121,698)(122,697)(123,696)(124,695)(125,694)(126,693)(127,692)(128,691)(129,690)(130,689)(131,688)(132,687)(133,686)(134,685)(135,684)(136,683)(137,682)(138,681)(139,680)(140,679)(141,678)(142,677)(143,676)(144,675)(145,674)(146,673)(147,672)(148,671)(149,670)(150,669)(151,668)(152,667)(153,666)(154,665)(155,664)(156,663)(157,662)(158,661)(159,660)(160,659)(161,658)(162,657)(163,656)(164,655)(165,654)(166,653)(167,652)(168,651)(169,650)(170,649)(171,648)(172,647)(173,646)(174,645)(175,644)(176,643)(177,642)(178,641)(179,640)(180,639)(181,638)(182,637)(183,636)(184,635)(185,634)(186,633)(187,632)(188,631)(189,630)(190,629)(191,628)(192,627)(193,626)(194,625)(195,624)(196,623)(197,622)(198,621)(199,620)(200,619)(201,618)(202,617)(203,616)(204,615)(205,614)(206,613)(207,612)(208,611)(209,610)(210,609)(211,608)(212,607)(213,606)(214,605)(215,604)(216,603)(217,602)(218,601)(219,600)(220,599)(221,598)(222,597)(223,596)(224,595)(225,594)(226,593)(227,592)(228,591)(229,590)(230,589)(231,588)(232,587)(233,586)(234,585)(235,584)(236,583)(237,582)(238,581)(239,580)(240,579)(241,578)(242,577)(243,576)(244,575)(245,574)(246,573)(247,572)(248,571)(249,570)(250,569)(251,568)(252,567)(253,566)(254,565)(255,564)(256,563)(257,562)(258,561)(259,560)(260,559)(261,558)(262,557)(263,556)(264,555)(265,554)(266,553)(267,552)(268,551)(269,550)(270,549)(271,548)(272,547)(273,546)(274,545)(275,544)(276,543)(277,542)(278,541)(279,540)(280,539)(281,538)(282,537)(283,536)(284,535)(285,534)(286,533)(287,532)(288,531)(289,530)(290,529)(291,528)(292,527)(293,526)(294,525)(295,524)(296,523)(297,522)(298,521)(299,520)(300,519)(301,518)(302,517)(303,516)(304,515)(305,514)(306,513)(307,512)(308,511)(309,510)(310,509)(311,508)(312,507)(313,506)(314,505)(315,504)(316,503)(317,502)(318,501)(319,500)(320,499)(321,498)(322,497)(323,496)(324,495)(325,494)(326,493)(327,492)(328,491)(329,490)(330,489)(331,488)(332,487)(333,486)(334,485)(335,484)(336,483)(337,482)(338,481)(339,480)(340,479)(341,478)(342,477)(343,476)(344,475)(345,474)(346,473)(347,472)(348,471)(349,470)(350,469)(351,468)(352,467)(353,466)(354,465)(355,464)(356,463)(357,462)(358,461)(359,460)(360,459)(361,458)(362,457)(363,456)(364,455)(365,454)(366,453)(367,452)(368,451)(369,450)(370,449)(371,448)(372,447)(373,446)(374,445)(375,444)(376,443)(377,442)(378,441)(379,440)(380,439)(381,438)(382,437)(383,436)(384,435)(385,434)(386,433)(387,432)(388,431)(389,430)(390,429)(391,428)(392,427)(393,426)(394,425)(395,424)(396,423)(397,422)(398,421)(399,420)(400,419)(401,418)(402,417)(403,416)(404,415)(405,414)(406,413)(407,412)(408,411)(409,410)

Conjugacy classes & character table

8373 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33474, 5);

C3×D5579 #45691049