D₂₆₀₁₆ #72289132

order	52032
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₀₁₆

Splittings

Fitting	C₃×C₃₂×C₂₇₁ ⋊ C₂
Hall	C₃×C₂₇₁ ⋊ D₃₂
Sylow	C₂₇₁ ⋊ D₉₆
Sylow	C₃ ⋊ D₈₆₇₂

Subgroups

	subgroup	G/H
center	C₂	D₁₃₀₀₈
commutator, cosocle	C₃×C₁₆×C₂₇₁	C₂²
Frattini	C₁₆	C₂×D₈₁₃
Fitting	C₃×C₃₂×C₂₇₁	C₂
socle	C₂×C₃×C₂₇₁	D₁₆
hypercenter	C₃₂	D₈₁₃
hypocenter	C₃×C₂₇₁	D₃₂
LCS	G ▷ C₃×C₁₆×C₂₇₁ ▷ C₃×C₈×C₂₇₁ ▷ C₃×C₄×C₂₇₁ ▷ C₂×C₃×C₂₇₁ ▷ C₃×C₂₇₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 306
transitive degree	μ_t(G) = 26016

A minimal faithful permutation representation — 2 generators on 306 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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,35)(37,306)(38,305)(39,304)(40,303)(41,302)(42,301)(43,300)(44,299)(45,298)(46,297)(47,296)(48,295)(49,294)(50,293)(51,292)(52,291)(53,290)(54,289)(55,288)(56,287)(57,286)(58,285)(59,284)(60,283)(61,282)(62,281)(63,280)(64,279)(65,278)(66,277)(67,276)(68,275)(69,274)(70,273)(71,272)(72,271)(73,270)(74,269)(75,268)(76,267)(77,266)(78,265)(79,264)(80,263)(81,262)(82,261)(83,260)(84,259)(85,258)(86,257)(87,256)(88,255)(89,254)(90,253)(91,252)(92,251)(93,250)(94,249)(95,248)(96,247)(97,246)(98,245)(99,244)(100,243)(101,242)(102,241)(103,240)(104,239)(105,238)(106,237)(107,236)(108,235)(109,234)(110,233)(111,232)(112,231)(113,230)(114,229)(115,228)(116,227)(117,226)(118,225)(119,224)(120,223)(121,222)(122,221)(123,220)(124,219)(125,218)(126,217)(127,216)(128,215)(129,214)(130,213)(131,212)(132,211)(133,210)(134,209)(135,208)(136,207)(137,206)(138,205)(139,204)(140,203)(141,202)(142,201)(143,200)(144,199)(145,198)(146,197)(147,196)(148,195)(149,194)(150,193)(151,192)(152,191)(153,190)(154,189)(155,188)(156,187)(157,186)(158,185)(159,184)(160,183)(161,182)(162,181)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)

Polycyclic presentation:

2^6,3,271|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2,h^a=h^270

Conjugacy classes & character table

13011 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", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 271]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^270);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26016 #72289132