D₃₂₇₂₀ #72304304

order	65440
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₇₂₀

Splittings

Fitting	C₅×C₁₆×C₄₀₉ ⋊ C₂
Hall	C₅×C₄₀₉ ⋊ D₁₆
Sylow	C₄₀₉ ⋊ D₈₀
Sylow	C₅ ⋊ D₆₅₄₄

Subgroups

	subgroup	G/H
center	C₂	D₁₆₃₆₀
commutator, cosocle	C₅×C₈×C₄₀₉	C₂²
Frattini	C₈	C₂×D₂₀₄₅
Fitting	C₅×C₁₆×C₄₀₉	C₂
socle	C₂×C₅×C₄₀₉	D₈
hypercenter	C₁₆	D₂₀₄₅
hypocenter	C₅×C₄₀₉	D₁₆
LCS	G ▷ C₅×C₈×C₄₀₉ ▷ C₄×C₅×C₄₀₉ ▷ C₂×C₅×C₄₀₉ ▷ C₅×C₄₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 430
transitive degree	μ_t(G) = 32720

A minimal faithful permutation representation — 2 generators on 430 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,21)(19,20)(23,430)(24,429)(25,428)(26,427)(27,426)(28,425)(29,424)(30,423)(31,422)(32,421)(33,420)(34,419)(35,418)(36,417)(37,416)(38,415)(39,414)(40,413)(41,412)(42,411)(43,410)(44,409)(45,408)(46,407)(47,406)(48,405)(49,404)(50,403)(51,402)(52,401)(53,400)(54,399)(55,398)(56,397)(57,396)(58,395)(59,394)(60,393)(61,392)(62,391)(63,390)(64,389)(65,388)(66,387)(67,386)(68,385)(69,384)(70,383)(71,382)(72,381)(73,380)(74,379)(75,378)(76,377)(77,376)(78,375)(79,374)(80,373)(81,372)(82,371)(83,370)(84,369)(85,368)(86,367)(87,366)(88,365)(89,364)(90,363)(91,362)(92,361)(93,360)(94,359)(95,358)(96,357)(97,356)(98,355)(99,354)(100,353)(101,352)(102,351)(103,350)(104,349)(105,348)(106,347)(107,346)(108,345)(109,344)(110,343)(111,342)(112,341)(113,340)(114,339)(115,338)(116,337)(117,336)(118,335)(119,334)(120,333)(121,332)(122,331)(123,330)(124,329)(125,328)(126,327)(127,326)(128,325)(129,324)(130,323)(131,322)(132,321)(133,320)(134,319)(135,318)(136,317)(137,316)(138,315)(139,314)(140,313)(141,312)(142,311)(143,310)(144,309)(145,308)(146,307)(147,306)(148,305)(149,304)(150,303)(151,302)(152,301)(153,300)(154,299)(155,298)(156,297)(157,296)(158,295)(159,294)(160,293)(161,292)(162,291)(163,290)(164,289)(165,288)(166,287)(167,286)(168,285)(169,284)(170,283)(171,282)(172,281)(173,280)(174,279)(175,278)(176,277)(177,276)(178,275)(179,274)(180,273)(181,272)(182,271)(183,270)(184,269)(185,268)(186,267)(187,266)(188,265)(189,264)(190,263)(191,262)(192,261)(193,260)(194,259)(195,258)(196,257)(197,256)(198,255)(199,254)(200,253)(201,252)(202,251)(203,250)(204,249)(205,248)(206,247)(207,246)(208,245)(209,244)(210,243)(211,242)(212,241)(213,240)(214,239)(215,238)(216,237)(217,236)(218,235)(219,234)(220,233)(221,232)(222,231)(223,230)(224,229)(225,228)(226,227)

Polycyclic presentation:

2^5,5,409|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^4,g^a=g^408

Conjugacy classes & character table

16363 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", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 5, 409]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
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);;
gap> SetConjugate(coll, 7, 1, F.7^408);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32720 #72304304