Dic₁₅₂₀₃ #72292742

order	60812
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₂₀₃

Splittings

derived, Hall	C₂₃×C₆₆₁ ⋊ C₄
Sylow	C₆₆₁ ⋊ Dic₂₃
Sylow	C₂₃ ⋊ Dic₆₆₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅₂₀₃
commutator, hypocenter	C₂₃×C₆₆₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₂₃×C₆₆₁	C₂
LCS	G ▷ C₂₃×C₆₆₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 688
transitive degree	μ_t(G) = 60812

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

Polycyclic presentation:

2^2,23,661|a^2=b,c^a=c^22,d^a=d^660

Conjugacy classes & character table

15206 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d");;
gap> coll := SingleCollector(F, [2, 2, 23, 661]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^22);;
gap> SetConjugate(coll, 4, 1, F.4^660);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15203 #72292742