D₂₉₀₀₀ #72292595

order	58000
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₉₀₀₀

Splittings

Fitting	C₈×C₂₉×C₁₂₅ ⋊ C₂
Hall	C₂₉×C₁₂₅ ⋊ D₈
Sylow	C₁₂₅ ⋊ D₂₃₂
Sylow	C₂₉ ⋊ D₁₀₀₀

Subgroups

	subgroup	G/H
center	C₂	D₁₄₅₀₀
commutator, cosocle	C₄×C₂₉×C₁₂₅	C₂²
Frattini	C₄×C₂₅	C₂×D₁₄₅
Fitting	C₈×C₂₉×C₁₂₅	C₂
socle	C₂×C₅×C₂₉	D₁₀₀
hypercenter	C₈	D₃₆₂₅
hypocenter	C₂₉×C₁₂₅	D₈
LCS	G ▷ C₄×C₂₉×C₁₂₅ ▷ C₂×C₂₉×C₁₂₅ ▷ C₂₉×C₁₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 162
transitive degree	μ_t(G) = 29000

A minimal faithful permutation representation — 2 generators on 162 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);(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)(135,162)(136,161)(137,160)(138,159)(139,158)(140,157)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)

Polycyclic presentation:

2^4,5,29,5^2|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^28,g^a=g^4h^4,h^a=h^4

Conjugacy classes & character table

14503 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, 29, 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.6^28);;
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);

D29000 #72292595