Dic₉₀₁₅ #49115134

order	36060
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₉₀₁₅
unambiguous semidirect product	C₃⋊Dic₃₀₀₅

Splittings

derived, Hall	C₃×C₅×C₆₀₁ ⋊ C₄
Hall	C₅×C₆₀₁ ⋊ Dic₃
Hall	C₃×C₆₀₁ ⋊ Dic₅
Sylow	C₆₀₁ ⋊ Dic₁₅
Hall	C₃×C₅ ⋊ Dic₆₀₁
Sylow	C₅ ⋊ Dic₁₈₀₃
Sylow	C₃ ⋊ Dic₃₀₀₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₉₀₁₅
commutator, hypocenter	C₃×C₅×C₆₀₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₆₀₁	C₂
LCS	G ▷ C₃×C₅×C₆₀₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 613
transitive degree	μ_t(G) = 36060

A minimal faithful permutation representation — 2 generators on 613 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);(1,3,2,4)(6,7)(9,12)(10,11)(14,613)(15,612)(16,611)(17,610)(18,609)(19,608)(20,607)(21,606)(22,605)(23,604)(24,603)(25,602)(26,601)(27,600)(28,599)(29,598)(30,597)(31,596)(32,595)(33,594)(34,593)(35,592)(36,591)(37,590)(38,589)(39,588)(40,587)(41,586)(42,585)(43,584)(44,583)(45,582)(46,581)(47,580)(48,579)(49,578)(50,577)(51,576)(52,575)(53,574)(54,573)(55,572)(56,571)(57,570)(58,569)(59,568)(60,567)(61,566)(62,565)(63,564)(64,563)(65,562)(66,561)(67,560)(68,559)(69,558)(70,557)(71,556)(72,555)(73,554)(74,553)(75,552)(76,551)(77,550)(78,549)(79,548)(80,547)(81,546)(82,545)(83,544)(84,543)(85,542)(86,541)(87,540)(88,539)(89,538)(90,537)(91,536)(92,535)(93,534)(94,533)(95,532)(96,531)(97,530)(98,529)(99,528)(100,527)(101,526)(102,525)(103,524)(104,523)(105,522)(106,521)(107,520)(108,519)(109,518)(110,517)(111,516)(112,515)(113,514)(114,513)(115,512)(116,511)(117,510)(118,509)(119,508)(120,507)(121,506)(122,505)(123,504)(124,503)(125,502)(126,501)(127,500)(128,499)(129,498)(130,497)(131,496)(132,495)(133,494)(134,493)(135,492)(136,491)(137,490)(138,489)(139,488)(140,487)(141,486)(142,485)(143,484)(144,483)(145,482)(146,481)(147,480)(148,479)(149,478)(150,477)(151,476)(152,475)(153,474)(154,473)(155,472)(156,471)(157,470)(158,469)(159,468)(160,467)(161,466)(162,465)(163,464)(164,463)(165,462)(166,461)(167,460)(168,459)(169,458)(170,457)(171,456)(172,455)(173,454)(174,453)(175,452)(176,451)(177,450)(178,449)(179,448)(180,447)(181,446)(182,445)(183,444)(184,443)(185,442)(186,441)(187,440)(188,439)(189,438)(190,437)(191,436)(192,435)(193,434)(194,433)(195,432)(196,431)(197,430)(198,429)(199,428)(200,427)(201,426)(202,425)(203,424)(204,423)(205,422)(206,421)(207,420)(208,419)(209,418)(210,417)(211,416)(212,415)(213,414)(214,413)(215,412)(216,411)(217,410)(218,409)(219,408)(220,407)(221,406)(222,405)(223,404)(224,403)(225,402)(226,401)(227,400)(228,399)(229,398)(230,397)(231,396)(232,395)(233,394)(234,393)(235,392)(236,391)(237,390)(238,389)(239,388)(240,387)(241,386)(242,385)(243,384)(244,383)(245,382)(246,381)(247,380)(248,379)(249,378)(250,377)(251,376)(252,375)(253,374)(254,373)(255,372)(256,371)(257,370)(258,369)(259,368)(260,367)(261,366)(262,365)(263,364)(264,363)(265,362)(266,361)(267,360)(268,359)(269,358)(270,357)(271,356)(272,355)(273,354)(274,353)(275,352)(276,351)(277,350)(278,349)(279,348)(280,347)(281,346)(282,345)(283,344)(284,343)(285,342)(286,341)(287,340)(288,339)(289,338)(290,337)(291,336)(292,335)(293,334)(294,333)(295,332)(296,331)(297,330)(298,329)(299,328)(300,327)(301,326)(302,325)(303,324)(304,323)(305,322)(306,321)(307,320)(308,319)(309,318)(310,317)(311,316)(312,315)(313,314)

Polycyclic presentation:

2,3,5,601,2|a^2=e,b^a=b^2,c^a=c^4,d^a=d^600

Conjugacy classes & character table

9018 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36060, 11);

Dic9015 #49115134