Dic₅₀₀₀ #72216596

order	20000
automorphism	Aut(G) order 40000000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₀₀₀

Splittings

Sylow

C₆₂₅ ⋊ Q₃₂

Subgroups

	subgroup	G/H
center	C₂	D₅₀₀₀
commutator, cosocle	C₈×C₆₂₅	C₂²
Frattini	C₈×C₁₂₅	C₂×D₅
Fitting	C₁₆×C₆₂₅	C₂
socle, prime radical	C₂×C₅	D₁₀₀₀
hypercenter	C₁₆	D₆₂₅
hypocenter	C₆₂₅	Q₃₂
LCS	G ▷ C₈×C₆₂₅ ▷ C₄×C₆₂₅ ▷ C₂×C₆₂₅ ▷ C₆₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 657
transitive degree	μ_t(G) = 20000

A minimal faithful permutation representation — 2 generators on 657 points:

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

Polycyclic presentation:

2^5,5^4|a^2=e,b^2=cd,c^2=de,d^2=e,f^5=g,g^5=h,h^5=i,b^a=bc,c^a=cd,d^a=de,f^a=f^4g^4h^4i^4,g^a=g^4h^4i^4,h^a=h^4i^4,i^a=i^4

Conjugacy classes & character table

5003 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₀₀₀, Dic₁₅₀₀₀
maximal normal subgroup of	Dic₁₀₀₀₀
maximal quotient of	Dic₁₅₀₀₀

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 5, 5, 5, 5]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^4*F.7^4*F.8^4*F.9^4);;
gap> SetConjugate(coll, 7, 1, F.7^4*F.8^4*F.9^4);;
gap> SetConjugate(coll, 8, 1, F.8^4*F.9^4);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5000 #72216596