D₇₄₆₄ #72212690

order	14928
automorphism	Aut(G) order 18510720
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₇₄₆₄

Splittings

Fitting	C₃×C₈×C₃₁₁ ⋊ C₂
Hall	C₃×C₃₁₁ ⋊ D₈
Sylow	C₃₁₁ ⋊ D₂₄
Sylow	C₃ ⋊ D₂₄₈₈

Subgroups

	subgroup	G/H
center	C₂	D₃₇₃₂
commutator, cosocle	C₃×C₄×C₃₁₁	C₂²
Frattini	C₄	C₂×D₉₃₃
Fitting	C₃×C₈×C₃₁₁	C₂
socle	C₂×C₃×C₃₁₁	D₄
hypercenter	C₈	D₉₃₃
hypocenter	C₃×C₃₁₁	D₈
LCS	G ▷ C₃×C₄×C₃₁₁ ▷ C₂×C₃×C₃₁₁ ▷ C₃×C₃₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 322
transitive degree	μ_t(G) = 7464

A minimal faithful permutation representation — 2 generators on 322 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);(2,8)(3,7)(4,6)(10,11)(13,322)(14,321)(15,320)(16,319)(17,318)(18,317)(19,316)(20,315)(21,314)(22,313)(23,312)(24,311)(25,310)(26,309)(27,308)(28,307)(29,306)(30,305)(31,304)(32,303)(33,302)(34,301)(35,300)(36,299)(37,298)(38,297)(39,296)(40,295)(41,294)(42,293)(43,292)(44,291)(45,290)(46,289)(47,288)(48,287)(49,286)(50,285)(51,284)(52,283)(53,282)(54,281)(55,280)(56,279)(57,278)(58,277)(59,276)(60,275)(61,274)(62,273)(63,272)(64,271)(65,270)(66,269)(67,268)(68,267)(69,266)(70,265)(71,264)(72,263)(73,262)(74,261)(75,260)(76,259)(77,258)(78,257)(79,256)(80,255)(81,254)(82,253)(83,252)(84,251)(85,250)(86,249)(87,248)(88,247)(89,246)(90,245)(91,244)(92,243)(93,242)(94,241)(95,240)(96,239)(97,238)(98,237)(99,236)(100,235)(101,234)(102,233)(103,232)(104,231)(105,230)(106,229)(107,228)(108,227)(109,226)(110,225)(111,224)(112,223)(113,222)(114,221)(115,220)(116,219)(117,218)(118,217)(119,216)(120,215)(121,214)(122,213)(123,212)(124,211)(125,210)(126,209)(127,208)(128,207)(129,206)(130,205)(131,204)(132,203)(133,202)(134,201)(135,200)(136,199)(137,198)(138,197)(139,196)(140,195)(141,194)(142,193)(143,192)(144,191)(145,190)(146,189)(147,188)(148,187)(149,186)(150,185)(151,184)(152,183)(153,182)(154,181)(155,180)(156,179)(157,178)(158,177)(159,176)(160,175)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)

Polycyclic presentation:

2^4,3,311|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^310

Conjugacy classes & character table

3735 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₄₉₂₈, D₂₂₃₉₂
maximal normal subgroup of	D₁₄₉₂₈
maximal quotient of	D₁₄₉₂₈, Dic₇₄₆₄, D₂₂₃₉₂

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 311]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^310);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D7464 #72212690