D₄₇₆₄ #72211488

order	9528
automorphism	Aut(G) order 7546176
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₄₇₆₄

Splittings

Fitting	C₃×C₄×C₃₉₇ ⋊ C₂
Hall	C₃×C₃₉₇ ⋊ D₄
Sylow	C₃₉₇ ⋊ D₁₂
Sylow	C₃ ⋊ D₁₅₈₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₁₉₁
commutator, socle, cosocle	C₂×C₃×C₃₉₇	C₂²
Fitting	C₃×C₄×C₃₉₇	C₂
hypercenter	C₄	D₁₁₉₁
hypocenter	C₃×C₃₉₇	D₄
LCS	G ▷ C₂×C₃×C₃₉₇ ▷ C₃×C₃₉₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 404
transitive degree	μ_t(G) = 4764

A minimal faithful permutation representation — 2 generators on 404 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);(2,4)(6,7)(9,404)(10,403)(11,402)(12,401)(13,400)(14,399)(15,398)(16,397)(17,396)(18,395)(19,394)(20,393)(21,392)(22,391)(23,390)(24,389)(25,388)(26,387)(27,386)(28,385)(29,384)(30,383)(31,382)(32,381)(33,380)(34,379)(35,378)(36,377)(37,376)(38,375)(39,374)(40,373)(41,372)(42,371)(43,370)(44,369)(45,368)(46,367)(47,366)(48,365)(49,364)(50,363)(51,362)(52,361)(53,360)(54,359)(55,358)(56,357)(57,356)(58,355)(59,354)(60,353)(61,352)(62,351)(63,350)(64,349)(65,348)(66,347)(67,346)(68,345)(69,344)(70,343)(71,342)(72,341)(73,340)(74,339)(75,338)(76,337)(77,336)(78,335)(79,334)(80,333)(81,332)(82,331)(83,330)(84,329)(85,328)(86,327)(87,326)(88,325)(89,324)(90,323)(91,322)(92,321)(93,320)(94,319)(95,318)(96,317)(97,316)(98,315)(99,314)(100,313)(101,312)(102,311)(103,310)(104,309)(105,308)(106,307)(107,306)(108,305)(109,304)(110,303)(111,302)(112,301)(113,300)(114,299)(115,298)(116,297)(117,296)(118,295)(119,294)(120,293)(121,292)(122,291)(123,290)(124,289)(125,288)(126,287)(127,286)(128,285)(129,284)(130,283)(131,282)(132,281)(133,280)(134,279)(135,278)(136,277)(137,276)(138,275)(139,274)(140,273)(141,272)(142,271)(143,270)(144,269)(145,268)(146,267)(147,266)(148,265)(149,264)(150,263)(151,262)(152,261)(153,260)(154,259)(155,258)(156,257)(157,256)(158,255)(159,254)(160,253)(161,252)(162,251)(163,250)(164,249)(165,248)(166,247)(167,246)(168,245)(169,244)(170,243)(171,242)(172,241)(173,240)(174,239)(175,238)(176,237)(177,236)(178,235)(179,234)(180,233)(181,232)(182,231)(183,230)(184,229)(185,228)(186,227)(187,226)(188,225)(189,224)(190,223)(191,222)(192,221)(193,220)(194,219)(195,218)(196,217)(197,216)(198,215)(199,214)(200,213)(201,212)(202,211)(203,210)(204,209)(205,208)(206,207)

Polycyclic presentation:

2^3,3,397|b^2=c,b^a=bc,d^a=d^2,e^a=e^396

Conjugacy classes & character table

2385 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉₅₂₈, D₁₄₂₉₂, D₂₃₈₂₀
maximal normal subgroup of	D₉₅₂₈
maximal quotient of	D₉₅₂₈, Dic₄₇₆₄, D₁₄₂₉₂, …

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, 3, 397]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^396);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D4764 #72211488