D₃₁₀₀₀ #72298574

order	62000
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₁₀₀₀

Splittings

Fitting	C₈×C₃₁×C₁₂₅ ⋊ C₂
Hall	C₃₁×C₁₂₅ ⋊ D₈
Sylow	C₁₂₅ ⋊ D₂₄₈
Sylow	C₃₁ ⋊ D₁₀₀₀

Subgroups

	subgroup	G/H
center	C₂	D₁₅₅₀₀
commutator, cosocle	C₄×C₃₁×C₁₂₅	C₂²
Frattini	C₄×C₂₅	C₂×D₁₅₅
Fitting	C₈×C₃₁×C₁₂₅	C₂
socle	C₂×C₅×C₃₁	D₁₀₀
hypercenter	C₈	D₃₈₇₅
hypocenter	C₃₁×C₁₂₅	D₈
LCS	G ▷ C₄×C₃₁×C₁₂₅ ▷ C₂×C₃₁×C₁₂₅ ▷ C₃₁×C₁₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 164
transitive degree	μ_t(G) = 31000

A minimal faithful permutation representation — 2 generators on 164 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);(2,8)(3,7)(4,6)(10,133)(11,132)(12,131)(13,130)(14,129)(15,128)(16,127)(17,126)(18,125)(19,124)(20,123)(21,122)(22,121)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(135,164)(136,163)(137,162)(138,161)(139,160)(140,159)(141,158)(142,157)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150)

Polycyclic presentation:

2^4,5,31,5^2|b^2=cd,c^2=d,e^5=g,g^5=h,b^a=bc,c^a=cd,e^a=e^4g^4h^4,f^a=f^30,g^a=g^4h^4,h^a=h^4

Conjugacy classes & character table

15503 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, 5, 31, 5, 5]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.7^4*F.8^4);;
gap> SetConjugate(coll, 6, 1, F.6^30);;
gap> SetConjugate(coll, 7, 1, F.7^4*F.8^4);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31000 #72298574