D₄₉₂₀ #72211547

order	9840
automorphism	Aut(G) order 6297600
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₄₉₂₀

Splittings

Fitting	C₃×C₅×C₈×C₄₁ ⋊ C₂
Hall	C₃×C₅×C₄₁ ⋊ D₈
Hall	C₅×C₄₁ ⋊ D₂₄
Hall	C₃×C₄₁ ⋊ D₄₀
Sylow	C₄₁ ⋊ D₁₂₀
Hall	C₃×C₅ ⋊ D₃₂₈
Sylow	C₅ ⋊ D₉₈₄
Sylow	C₃ ⋊ D₁₆₄₀

Subgroups

	subgroup	G/H
center	C₂	D₂₄₆₀
commutator, cosocle	C₃×C₄×C₅×C₄₁	C₂²
Frattini	C₄	C₂×D₆₁₅
Fitting	C₃×C₅×C₈×C₄₁	C₂
socle	C₂×C₃×C₅×C₄₁	D₄
hypercenter	C₈	D₆₁₅
hypocenter	C₃×C₅×C₄₁	D₈
LCS	G ▷ C₃×C₄×C₅×C₄₁ ▷ C₂×C₃×C₅×C₄₁ ▷ C₃×C₅×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 4920

A minimal faithful permutation representation — 2 generators on 57 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);(2,8)(3,7)(4,6)(10,11)(13,16)(14,15)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)

Polycyclic presentation:

2^4,3,5,41|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^40

Conjugacy classes & character table

2463 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉₈₄₀, D₁₄₇₆₀, D₂₄₆₀₀
maximal normal subgroup of	D₉₈₄₀
maximal quotient of	D₉₈₄₀, Dic₄₉₂₀, D₁₄₇₆₀, …

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, 2, 3, 5, 41]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D4920 #72211547