D₃₂₆₁₂ #72303663

order	65224
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₆₁₂

Splittings

Fitting	C₄×C₃₁×C₂₆₃ ⋊ C₂
Hall	C₃₁×C₂₆₃ ⋊ D₄
Sylow	C₂₆₃ ⋊ D₁₂₄
Sylow	C₃₁ ⋊ D₁₀₅₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₈₁₅₃
commutator, socle, cosocle	C₂×C₃₁×C₂₆₃	C₂²
Fitting	C₄×C₃₁×C₂₆₃	C₂
hypercenter	C₄	D₈₁₅₃
hypocenter	C₃₁×C₂₆₃	D₄
LCS	G ▷ C₂×C₃₁×C₂₆₃ ▷ C₃₁×C₂₆₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 298
transitive degree	μ_t(G) = 32612

A minimal faithful permutation representation — 2 generators on 298 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);(2,4)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(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:

2^3,31,263|b^2=c,b^a=bc,d^a=d^30,e^a=e^262

Conjugacy classes & character table

16309 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 31, 263]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^30);;
gap> SetConjugate(coll, 5, 1, F.5^262);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32612 #72303663