D₂₅₅₀₀ #72288653

order	51000
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₅₅₀₀

Splittings

Fitting	C₃×C₄×C₁₇×C₁₂₅ ⋊ C₂
Hall	C₃×C₁₇×C₁₂₅ ⋊ D₄
Hall	C₁₇×C₁₂₅ ⋊ D₁₂
Hall	C₃×C₁₂₅ ⋊ D₆₈
Sylow	C₁₂₅ ⋊ D₂₀₄
Hall	C₃×C₁₇ ⋊ D₅₀₀
Sylow	C₁₇ ⋊ D₁₅₀₀
Sylow	C₃ ⋊ D₈₅₀₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₆₃₇₅
commutator, cosocle	C₂×C₃×C₁₇×C₁₂₅	C₂²
Frattini	C₂×C₂₅	C₂×D₂₅₅
Fitting	C₃×C₄×C₁₇×C₁₂₅	C₂
socle	C₂×C₃×C₅×C₁₇	C₂×D₂₅
hypercenter	C₄	D₆₃₇₅
hypocenter	C₃×C₁₇×C₁₂₅	D₄
LCS	G ▷ C₂×C₃×C₁₇×C₁₂₅ ▷ C₃×C₁₇×C₁₂₅
UCS	C₁ ◁ C₂ ◁ C₄

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 149
transitive degree	μ_t(G) = 25500

A minimal faithful permutation representation — 2 generators on 149 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);(2,4)(6,7)(9,132)(10,131)(11,130)(12,129)(13,128)(14,127)(15,126)(16,125)(17,124)(18,123)(19,122)(20,121)(21,120)(22,119)(23,118)(24,117)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)(141,142)

Polycyclic presentation:

2^3,3,5,17,5^2|b^2=c,e^5=g,g^5=h,b^a=bc,d^a=d^2,e^a=e^4g^4h^4,f^a=f^16,g^a=g^4h^4,h^a=h^4

Conjugacy classes & character table

12753 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, 3, 5, 17, 5, 5]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.7^4*F.8^4);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
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);

D25500 #72288653