C₅×D₃×D₅₂₉ #44540677

order	31740
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×D₃×D₅₂₉
unambiguous semidirect product	(D₃×C₅₂₉)⋊(C₂×C₅)
unambiguous semidirect product	(C₃×D₅₂₉)⋊(C₂×C₅)
unambiguous semidirect product	D₁₅₈₇⋊(C₂×C₅)

Splittings

Fitting, Hall	C₃×C₅×C₅₂₉ ⋊ C₂²
derived, Hall	C₃×C₅₂₉ ⋊ C₂²×C₅
Hall	C₅×C₅₂₉ ⋊ C₂×D₃
Sylow	C₅₂₉ ⋊ C₂×C₅×D₃
Hall	C₃×C₅ ⋊ C₂×D₅₂₉
Sylow	C₃ ⋊ C₂×C₅×D₅₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₃×D₅₂₉
commutator, cosocle, hypocenter	C₃×C₅₂₉	C₂²×C₅
Frattini	C₂₃	C₅×D₃×D₂₃
Fitting	C₃×C₅×C₅₂₉	C₂²
socle	C₃×C₅×C₂₃	C₂×D₂₃
LCS	G ▷ C₃×C₅₂₉
UCS	C₁ ◁ C₅

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 537
transitive degree	μ_t(G) = 7935

A minimal faithful permutation representation — 2 generators on 537 points:

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

Conjugacy classes & character table

3990 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31740, 15);

C5×D3×D529 #44540677