D₁₆₀₄₃ #44545549

order	32086
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₀₄₃
unambiguous semidirect product	C₆₁⋊D₂₆₃
unambiguous semidirect product	C₂₆₃⋊D₆₁

Splittings

Fitting, socle, derived, Hall	C₆₁×C₂₆₃ ⋊ C₂
Sylow	C₂₆₃ ⋊ D₆₁
Sylow	C₆₁ ⋊ D₂₆₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₆₁×C₂₆₃	C₂
LCS	G ▷ C₆₁×C₂₆₃
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 324
transitive degree	μ_t(G) = 16043

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);(2,61)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(63,324)(64,323)(65,322)(66,321)(67,320)(68,319)(69,318)(70,317)(71,316)(72,315)(73,314)(74,313)(75,312)(76,311)(77,310)(78,309)(79,308)(80,307)(81,306)(82,305)(83,304)(84,303)(85,302)(86,301)(87,300)(88,299)(89,298)(90,297)(91,296)(92,295)(93,294)(94,293)(95,292)(96,291)(97,290)(98,289)(99,288)(100,287)(101,286)(102,285)(103,284)(104,283)(105,282)(106,281)(107,280)(108,279)(109,278)(110,277)(111,276)(112,275)(113,274)(114,273)(115,272)(116,271)(117,270)(118,269)(119,268)(120,267)(121,266)(122,265)(123,264)(124,263)(125,262)(126,261)(127,260)(128,259)(129,258)(130,257)(131,256)(132,255)(133,254)(134,253)(135,252)(136,251)(137,250)(138,249)(139,248)(140,247)(141,246)(142,245)(143,244)(144,243)(145,242)(146,241)(147,240)(148,239)(149,238)(150,237)(151,236)(152,235)(153,234)(154,233)(155,232)(156,231)(157,230)(158,229)(159,228)(160,227)(161,226)(162,225)(163,224)(164,223)(165,222)(166,221)(167,220)(168,219)(169,218)(170,217)(171,216)(172,215)(173,214)(174,213)(175,212)(176,211)(177,210)(178,209)(179,208)(180,207)(181,206)(182,205)(183,204)(184,203)(185,202)(186,201)(187,200)(188,199)(189,198)(190,197)(191,196)(192,195)(193,194)

Polycyclic presentation:

2,61,263|b^a=b^60,c^a=c^262

Conjugacy classes & character table

8023 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₆₀₄₃
maximal normal subgroup of	C₂×D₁₆₀₄₃
maximal quotient of	Dic₁₆₀₄₃, C₂×D₁₆₀₄₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32086, 3);

D16043 #44545549