D₁₈₁₈₀ #72273458

order	36360
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₈₁₈₀

Splittings

Fitting	C₄×C₅×C₉×C₁₀₁ ⋊ C₂
Hall	C₅×C₉×C₁₀₁ ⋊ D₄
Hall	C₉×C₁₀₁ ⋊ D₂₀
Hall	C₅×C₁₀₁ ⋊ D₃₆
Sylow	C₁₀₁ ⋊ D₁₈₀
Hall	C₅×C₉ ⋊ D₄₀₄
Sylow	C₉ ⋊ D₂₀₂₀
Sylow	C₅ ⋊ D₃₆₃₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄₅₄₅
commutator, cosocle	C₂×C₅×C₉×C₁₀₁	C₂²
Frattini	C₂×C₃	C₂×D₁₅₁₅
Fitting	C₄×C₅×C₉×C₁₀₁	C₂
socle	C₂×C₃×C₅×C₁₀₁	C₂×D₃
hypercenter	C₄	D₄₅₄₅
hypocenter	C₅×C₉×C₁₀₁	D₄
LCS	G ▷ C₂×C₅×C₉×C₁₀₁ ▷ C₅×C₉×C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 18180

A minimal faithful permutation representation — 2 generators on 119 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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,18)(16,17)(20,119)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,112)(28,111)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Polycyclic presentation:

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

Conjugacy classes & character table

9093 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, 101, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^100);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D18180 #72273458