D₉₂₆₀ #72214101

order	18520
automorphism	Aut(G) order 34224960
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₉₂₆₀

Splittings

Fitting	C₄×C₅×C₄₆₃ ⋊ C₂
Hall	C₅×C₄₆₃ ⋊ D₄
Sylow	C₄₆₃ ⋊ D₂₀
Sylow	C₅ ⋊ D₁₈₅₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₃₁₅
commutator, socle, cosocle	C₂×C₅×C₄₆₃	C₂²
Fitting	C₄×C₅×C₄₆₃	C₂
hypercenter	C₄	D₂₃₁₅
hypocenter	C₅×C₄₆₃	D₄
LCS	G ▷ C₂×C₅×C₄₆₃ ▷ C₅×C₄₆₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 472
transitive degree	μ_t(G) = 9260

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

Polycyclic presentation:

2^3,5,463|b^2=c,b^a=bc,d^a=d^4,e^a=e^462

Conjugacy classes & character table

4633 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₈₅₂₀, D₂₇₇₈₀
maximal normal subgroup of	D₁₈₅₂₀
maximal quotient of	Dic₉₂₆₀, D₁₈₅₂₀, D₂₇₇₈₀

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, 463]);;
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.5^462);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D9260 #72214101