Dic₅₀₃₀ #72216615

order	20120
automorphism	Aut(G) order 40400960
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₀₃₀

Splittings

Hall	C₅×C₅₀₃ ⋊ Q₈
Sylow	C₅₀₃ ⋊ Dic₁₀
Sylow	C₅ ⋊ Dic₁₀₀₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₅₁₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₅₀₃	C₂²
Fitting	C₄×C₅×C₅₀₃	C₂
hypercenter	C₄	D₂₅₁₅
hypocenter	C₅×C₅₀₃	Q₈
LCS	G ▷ C₂×C₅×C₅₀₃ ▷ C₅×C₅₀₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 516
transitive degree	μ_t(G) = 20120

A minimal faithful permutation representation — 2 generators on 516 points:

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

Polycyclic presentation:

2^3,5,503|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^502

Conjugacy classes & character table

5033 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₀₆₀, Dic₁₅₀₉₀
maximal normal subgroup of	Dic₁₀₀₆₀
maximal quotient of	Dic₁₅₀₉₀

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 503]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^502);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5030 #72216615