D₂₀₁₀₀ #72274287

order	40200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₀₁₀₀

Splittings

Fitting	C₃×C₄×C₂₅×C₆₇ ⋊ C₂
Hall	C₃×C₂₅×C₆₇ ⋊ D₄
Hall	C₂₅×C₆₇ ⋊ D₁₂
Hall	C₃×C₆₇ ⋊ D₁₀₀
Hall	C₃×C₂₅ ⋊ D₂₆₈
Sylow	C₆₇ ⋊ D₃₀₀
Sylow	C₂₅ ⋊ D₈₀₄
Sylow	C₃ ⋊ D₆₇₀₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₅₀₂₅
commutator, cosocle	C₂×C₃×C₂₅×C₆₇	C₂²
Frattini	C₂×C₅	C₂×D₁₀₀₅
Fitting	C₃×C₄×C₂₅×C₆₇	C₂
socle	C₂×C₃×C₅×C₆₇	C₂×D₅
hypercenter	C₄	D₅₀₂₅
hypocenter	C₃×C₂₅×C₆₇	D₄
LCS	G ▷ C₂×C₃×C₂₅×C₆₇ ▷ C₃×C₂₅×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 20100

A minimal faithful permutation representation — 2 generators on 99 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);(2,4)(6,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(34,99)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Polycyclic presentation:

2^3,3,5,67,5|b^2=c,e^5=g,b^a=bc,d^a=d^2,e^a=e^4g^4,f^a=f^66,g^a=g^4

Conjugacy classes & character table

10053 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 67, 5]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.7);;
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);;
gap> SetConjugate(coll, 6, 1, F.6^66);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D20100 #72274287