D₆₀₆₀ #72212033

order	12120
automorphism	Aut(G) order 9696000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₆₀₆₀

Splittings

Fitting	C₃×C₄×C₅×C₁₀₁ ⋊ C₂
Hall	C₃×C₅×C₁₀₁ ⋊ D₄
Hall	C₅×C₁₀₁ ⋊ D₁₂
Hall	C₃×C₁₀₁ ⋊ D₂₀
Sylow	C₁₀₁ ⋊ D₆₀
Hall	C₃×C₅ ⋊ D₄₀₄
Sylow	C₅ ⋊ D₁₂₁₂
Sylow	C₃ ⋊ D₂₀₂₀

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₅₁₅
commutator, socle, cosocle	C₂×C₃×C₅×C₁₀₁	C₂²
Fitting	C₃×C₄×C₅×C₁₀₁	C₂
hypercenter	C₄	D₁₅₁₅
hypocenter	C₃×C₅×C₁₀₁	D₄
LCS	G ▷ C₂×C₃×C₅×C₁₀₁ ▷ C₃×C₅×C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 6060

A minimal faithful permutation representation — 2 generators on 113 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);(2,4)(6,7)(9,12)(10,11)(14,113)(15,112)(16,111)(17,110)(18,109)(19,108)(20,107)(21,106)(22,105)(23,104)(24,103)(25,102)(26,101)(27,100)(28,99)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2^3,3,5,101|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^100

Conjugacy classes & character table

3033 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₂₁₂₀, D₁₈₁₈₀, D₃₀₃₀₀
maximal normal subgroup of	D₁₂₁₂₀
maximal quotient of	D₁₂₁₂₀, Dic₆₀₆₀, D₁₈₁₈₀, …

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 101]);;
gap> SetPower(coll, 2, F.3);;
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);;
gap> SetConjugate(coll, 6, 1, F.6^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D6060 #72212033