D₃₁₈₄₀ #72301772

order	63680
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₁₈₄₀

Splittings

Fitting	C₅×C₃₂×C₁₉₉ ⋊ C₂
Hall	C₅×C₁₉₉ ⋊ D₃₂
Sylow	C₁₉₉ ⋊ D₁₆₀
Sylow	C₅ ⋊ D₆₃₆₈

Subgroups

	subgroup	G/H
center	C₂	D₁₅₉₂₀
commutator, cosocle	C₅×C₁₆×C₁₉₉	C₂²
Frattini	C₁₆	C₂×D₉₉₅
Fitting	C₅×C₃₂×C₁₉₉	C₂
socle	C₂×C₅×C₁₉₉	D₁₆
hypercenter	C₃₂	D₉₉₅
hypocenter	C₅×C₁₉₉	D₃₂
LCS	G ▷ C₅×C₁₆×C₁₉₉ ▷ C₅×C₈×C₁₉₉ ▷ C₄×C₅×C₁₉₉ ▷ C₂×C₅×C₁₉₉ ▷ C₅×C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 236
transitive degree	μ_t(G) = 31840

A minimal faithful permutation representation — 2 generators on 236 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);(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)(34,37)(35,36)(39,236)(40,235)(41,234)(42,233)(43,232)(44,231)(45,230)(46,229)(47,228)(48,227)(49,226)(50,225)(51,224)(52,223)(53,222)(54,221)(55,220)(56,219)(57,218)(58,217)(59,216)(60,215)(61,214)(62,213)(63,212)(64,211)(65,210)(66,209)(67,208)(68,207)(69,206)(70,205)(71,204)(72,203)(73,202)(74,201)(75,200)(76,199)(77,198)(78,197)(79,196)(80,195)(81,194)(82,193)(83,192)(84,191)(85,190)(86,189)(87,188)(88,187)(89,186)(90,185)(91,184)(92,183)(93,182)(94,181)(95,180)(96,179)(97,178)(98,177)(99,176)(100,175)(101,174)(102,173)(103,172)(104,171)(105,170)(106,169)(107,168)(108,167)(109,166)(110,165)(111,164)(112,163)(113,162)(114,161)(115,160)(116,159)(117,158)(118,157)(119,156)(120,155)(121,154)(122,153)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)(134,141)(135,140)(136,139)(137,138)

Polycyclic presentation:

2^6,5,199|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^4,h^a=h^198

Conjugacy classes & character table

15923 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, 5, 199]);;
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.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^198);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31840 #72301772