D₈₇₆₄ #72213655

order	17528
automorphism	Aut(G) order 32812416
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₇₆₄

Splittings

Fitting	C₄×C₇×C₃₁₃ ⋊ C₂
Hall	C₇×C₃₁₃ ⋊ D₄
Sylow	C₃₁₃ ⋊ D₂₈
Sylow	C₇ ⋊ D₁₂₅₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₁₉₁
commutator, socle, cosocle	C₂×C₇×C₃₁₃	C₂²
Fitting	C₄×C₇×C₃₁₃	C₂
hypercenter	C₄	D₂₁₉₁
hypocenter	C₇×C₃₁₃	D₄
LCS	G ▷ C₂×C₇×C₃₁₃ ▷ C₇×C₃₁₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 324
transitive degree	μ_t(G) = 8764

A minimal faithful permutation representation — 2 generators on 324 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);(2,4)(6,11)(7,10)(8,9)(13,324)(14,323)(15,322)(16,321)(17,320)(18,319)(19,318)(20,317)(21,316)(22,315)(23,314)(24,313)(25,312)(26,311)(27,310)(28,309)(29,308)(30,307)(31,306)(32,305)(33,304)(34,303)(35,302)(36,301)(37,300)(38,299)(39,298)(40,297)(41,296)(42,295)(43,294)(44,293)(45,292)(46,291)(47,290)(48,289)(49,288)(50,287)(51,286)(52,285)(53,284)(54,283)(55,282)(56,281)(57,280)(58,279)(59,278)(60,277)(61,276)(62,275)(63,274)(64,273)(65,272)(66,271)(67,270)(68,269)(69,268)(70,267)(71,266)(72,265)(73,264)(74,263)(75,262)(76,261)(77,260)(78,259)(79,258)(80,257)(81,256)(82,255)(83,254)(84,253)(85,252)(86,251)(87,250)(88,249)(89,248)(90,247)(91,246)(92,245)(93,244)(94,243)(95,242)(96,241)(97,240)(98,239)(99,238)(100,237)(101,236)(102,235)(103,234)(104,233)(105,232)(106,231)(107,230)(108,229)(109,228)(110,227)(111,226)(112,225)(113,224)(114,223)(115,222)(116,221)(117,220)(118,219)(119,218)(120,217)(121,216)(122,215)(123,214)(124,213)(125,212)(126,211)(127,210)(128,209)(129,208)(130,207)(131,206)(132,205)(133,204)(134,203)(135,202)(136,201)(137,200)(138,199)(139,198)(140,197)(141,196)(142,195)(143,194)(144,193)(145,192)(146,191)(147,190)(148,189)(149,188)(150,187)(151,186)(152,185)(153,184)(154,183)(155,182)(156,181)(157,180)(158,179)(159,178)(160,177)(161,176)(162,175)(163,174)(164,173)(165,172)(166,171)(167,170)(168,169)

Polycyclic presentation:

2^3,7,313|b^2=c,b^a=bc,d^a=d^6,e^a=e^312

Conjugacy classes & character table

4385 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₅₂₈, D₂₆₂₉₂
maximal normal subgroup of	D₁₇₅₂₈
maximal quotient of	Dic₈₇₆₄, D₁₇₅₂₈, D₂₆₂₉₂

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, 7, 313]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^312);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8764 #72213655