D₁₁₅₀₀ #72215992

order	23000
automorphism	Aut(G) order 50600000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₅₀₀

Splittings

Fitting	C₄×C₂₃×C₁₂₅ ⋊ C₂
Hall	C₂₃×C₁₂₅ ⋊ D₄
Sylow	C₁₂₅ ⋊ D₉₂
Sylow	C₂₃ ⋊ D₅₀₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₈₇₅
commutator, cosocle	C₂×C₂₃×C₁₂₅	C₂²
Frattini	C₂×C₂₅	C₂×D₁₁₅
Fitting	C₄×C₂₃×C₁₂₅	C₂
socle	C₂×C₅×C₂₃	C₂×D₂₅
hypercenter	C₄	D₂₈₇₅
hypocenter	C₂₃×C₁₂₅	D₄
LCS	G ▷ C₂×C₂₃×C₁₂₅ ▷ C₂₃×C₁₂₅
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 152
transitive degree	μ_t(G) = 11500

A minimal faithful permutation representation — 2 generators on 152 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);(2,4)(6,129)(7,128)(8,127)(9,126)(10,125)(11,124)(12,123)(13,122)(14,121)(15,120)(16,119)(17,118)(18,117)(19,116)(20,115)(21,114)(22,113)(23,112)(24,111)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(131,152)(132,151)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)(141,142)

Polycyclic presentation:

2^3,5,23,5^2|b^2=c,d^5=f,f^5=g,b^a=bc,d^a=d^4f^4g^4,e^a=e^22,f^a=f^4g^4,g^a=g^4

Conjugacy classes & character table

5753 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₀₀₀
maximal normal subgroup of	D₂₃₀₀₀
maximal quotient of	Dic₁₁₅₀₀, D₂₃₀₀₀

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 23, 5, 5]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.6^4*F.7^4);;
gap> SetConjugate(coll, 5, 1, F.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^4*F.7^4);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11500 #72215992