D₁₁₆₆₀ #72223933

order	23320
automorphism	Aut(G) order 48505600
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₆₆₀

Splittings

Fitting	C₄×C₅×C₁₁×C₅₃ ⋊ C₂
Hall	C₅×C₁₁×C₅₃ ⋊ D₄
Hall	C₁₁×C₅₃ ⋊ D₂₀
Hall	C₅×C₅₃ ⋊ D₄₄
Hall	C₅×C₁₁ ⋊ D₂₁₂
Sylow	C₅₃ ⋊ D₂₂₀
Sylow	C₁₁ ⋊ D₁₀₆₀
Sylow	C₅ ⋊ D₂₃₃₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₉₁₅
commutator, socle, cosocle	C₂×C₅×C₁₁×C₅₃	C₂²
Fitting	C₄×C₅×C₁₁×C₅₃	C₂
hypercenter	C₄	D₂₉₁₅
hypocenter	C₅×C₁₁×C₅₃	D₄
LCS	G ▷ C₂×C₅×C₁₁×C₅₃ ▷ C₅×C₁₁×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 11660

A minimal faithful permutation representation — 2 generators on 73 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);(2,4)(6,9)(7,8)(11,20)(12,19)(13,18)(14,17)(15,16)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2^3,5,11,53|b^2=c,b^a=bc,d^a=d^4,e^a=e^10,f^a=f^52

Conjugacy classes & character table

5833 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₃₂₀
maximal normal subgroup of	D₂₃₃₂₀
maximal quotient of	Dic₁₁₆₆₀, D₂₃₃₂₀

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, 5, 11, 53]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11660 #72223933