C₂×C₅×D₁₅₈₇ #44540688

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

Aliases

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

Splittings

Fitting	C₂×C₃×C₅×C₅₂₉ ⋊ C₂
derived, Hall	C₃×C₅₂₉ ⋊ C₂²×C₅
Hall	C₃×C₅×C₅₂₉ ⋊ C₂²
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	C₂×C₅	D₁₅₈₇
commutator, cosocle, hypocenter	C₃×C₅₂₉	C₂²×C₅
Frattini	C₂₃	C₂×C₅×D₆₉
Fitting	C₂×C₃×C₅×C₅₂₉	C₂
socle	C₂×C₃×C₅×C₂₃	D₂₃
LCS	G ▷ C₃×C₅₂₉
UCS	C₁ ◁ C₂×C₅

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 539
transitive degree	μ_t(G) = 15870

A minimal faithful permutation representation — 2 generators on 539 points:

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

Conjugacy classes & character table

7950 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31740, 26);

C2×C5×D1587 #44540688