D₁₂₃₂₀ #72224082

order	24640
automorphism	Aut(G) order 47308800
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₂₃₂₀

Splittings

Fitting	C₅×C₇×C₁₁×C₃₂ ⋊ C₂
Hall	C₅×C₇×C₁₁ ⋊ D₃₂
Hall	C₇×C₁₁ ⋊ D₁₆₀
Hall	C₅×C₁₁ ⋊ D₂₂₄
Hall	C₅×C₇ ⋊ D₃₅₂
Sylow	C₁₁ ⋊ D₁₁₂₀
Sylow	C₇ ⋊ D₁₇₆₀
Sylow	C₅ ⋊ D₂₄₆₄

Subgroups

	subgroup	G/H
center	C₂	D₆₁₆₀
commutator, cosocle	C₅×C₇×C₁₁×C₁₆	C₂²
Frattini	C₁₆	C₂×D₃₈₅
Fitting	C₅×C₇×C₁₁×C₃₂	C₂
socle	C₂×C₅×C₇×C₁₁	D₁₆
hypercenter	C₃₂	D₃₈₅
hypocenter	C₅×C₇×C₁₁	D₃₂
LCS	G ▷ C₅×C₇×C₁₁×C₁₆ ▷ C₅×C₇×C₈×C₁₁ ▷ C₄×C₅×C₇×C₁₁ ▷ C₂×C₅×C₇×C₁₁ ▷ C₅×C₇×C₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 12320

A minimal faithful permutation representation — 2 generators on 55 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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,37)(35,36)(39,44)(40,43)(41,42)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2^6,5,7,11|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^4,h^a=h^6,i^a=i^10

Conjugacy classes & character table

6163 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₄₆₄₀
maximal normal subgroup of	D₂₄₆₄₀
maximal quotient of	Dic₁₂₃₂₀, D₂₄₆₄₀

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 5, 7, 11]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 9, 1, F.9^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D12320 #72224082