D₂₅₇₆₇ #72288902

order	51534
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₅₇₆₇

Splittings

Fitting, derived, Hall	C₇×C₉×C₄₀₉ ⋊ C₂
Hall	C₉×C₄₀₉ ⋊ D₇
Hall	C₇×C₄₀₉ ⋊ D₉
Sylow	C₄₀₉ ⋊ D₆₃
Hall	C₇×C₉ ⋊ D₄₀₉
Sylow	C₉ ⋊ D₂₈₆₃
Sylow	C₇ ⋊ D₃₆₈₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₉×C₄₀₉	C₂
Frattini	C₃	D₈₅₈₉
socle	C₃×C₇×C₄₀₉	D₃
LCS	G ▷ C₇×C₉×C₄₀₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 425
transitive degree	μ_t(G) = 25767

A minimal faithful permutation representation — 2 generators on 425 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);(2,9)(3,8)(4,7)(5,6)(11,16)(12,15)(13,14)(18,425)(19,424)(20,423)(21,422)(22,421)(23,420)(24,419)(25,418)(26,417)(27,416)(28,415)(29,414)(30,413)(31,412)(32,411)(33,410)(34,409)(35,408)(36,407)(37,406)(38,405)(39,404)(40,403)(41,402)(42,401)(43,400)(44,399)(45,398)(46,397)(47,396)(48,395)(49,394)(50,393)(51,392)(52,391)(53,390)(54,389)(55,388)(56,387)(57,386)(58,385)(59,384)(60,383)(61,382)(62,381)(63,380)(64,379)(65,378)(66,377)(67,376)(68,375)(69,374)(70,373)(71,372)(72,371)(73,370)(74,369)(75,368)(76,367)(77,366)(78,365)(79,364)(80,363)(81,362)(82,361)(83,360)(84,359)(85,358)(86,357)(87,356)(88,355)(89,354)(90,353)(91,352)(92,351)(93,350)(94,349)(95,348)(96,347)(97,346)(98,345)(99,344)(100,343)(101,342)(102,341)(103,340)(104,339)(105,338)(106,337)(107,336)(108,335)(109,334)(110,333)(111,332)(112,331)(113,330)(114,329)(115,328)(116,327)(117,326)(118,325)(119,324)(120,323)(121,322)(122,321)(123,320)(124,319)(125,318)(126,317)(127,316)(128,315)(129,314)(130,313)(131,312)(132,311)(133,310)(134,309)(135,308)(136,307)(137,306)(138,305)(139,304)(140,303)(141,302)(142,301)(143,300)(144,299)(145,298)(146,297)(147,296)(148,295)(149,294)(150,293)(151,292)(152,291)(153,290)(154,289)(155,288)(156,287)(157,286)(158,285)(159,284)(160,283)(161,282)(162,281)(163,280)(164,279)(165,278)(166,277)(167,276)(168,275)(169,274)(170,273)(171,272)(172,271)(173,270)(174,269)(175,268)(176,267)(177,266)(178,265)(179,264)(180,263)(181,262)(182,261)(183,260)(184,259)(185,258)(186,257)(187,256)(188,255)(189,254)(190,253)(191,252)(192,251)(193,250)(194,249)(195,248)(196,247)(197,246)(198,245)(199,244)(200,243)(201,242)(202,241)(203,240)(204,239)(205,238)(206,237)(207,236)(208,235)(209,234)(210,233)(211,232)(212,231)(213,230)(214,229)(215,228)(216,227)(217,226)(218,225)(219,224)(220,223)(221,222)

Polycyclic presentation:

2,3,7,409,3|b^3=e,b^a=b^2e^2,c^a=c^6,d^a=d^408,e^a=e^2

Conjugacy classes & character table

12885 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");;
gap> coll := SingleCollector(F, [2, 3, 7, 409, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^408);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D25767 #72288902