D₁₀₀₈₀ #72215198

order	20160
automorphism	Aut(G) order 23224320
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₀₈₀

Splittings

Fitting	C₅×C₇×C₉×C₃₂ ⋊ C₂
Hall	C₅×C₇×C₉ ⋊ D₃₂
Hall	C₇×C₉ ⋊ D₁₆₀
Hall	C₅×C₉ ⋊ D₂₂₄
Hall	C₅×C₇ ⋊ D₂₈₈
Sylow	C₉ ⋊ D₁₁₂₀
Sylow	C₇ ⋊ D₁₄₄₀
Sylow	C₅ ⋊ D₂₀₁₆

Subgroups

	subgroup	G/H
center	C₂	D₅₀₄₀
commutator, cosocle	C₅×C₇×C₉×C₁₆	C₂²
Frattini	C₃×C₁₆	C₂×D₁₀₅
Fitting	C₅×C₇×C₉×C₃₂	C₂
socle	C₂×C₃×C₅×C₇	D₄₈
hypercenter	C₃₂	D₃₁₅
hypocenter	C₅×C₇×C₉	D₃₂
LCS	G ▷ C₅×C₇×C₉×C₁₆ ▷ C₅×C₇×C₈×C₉ ▷ C₄×C₅×C₇×C₉ ▷ C₂×C₅×C₇×C₉ ▷ C₅×C₇×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

204 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 10080

A minimal faithful permutation representation — 2 generators on 53 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);(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)(34,41)(35,40)(36,39)(37,38)(43,46)(44,45)(48,53)(49,52)(50,51)

Polycyclic presentation:

2^6,3,5,7,3|b^2=cd,c^2=de,d^2=ef,e^2=f,g^3=j,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2j^2,h^a=h^4,i^a=i^6,j^a=j^2

Conjugacy classes & character table

5043 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₀₁₆₀, D₃₀₂₄₀
maximal normal subgroup of	D₂₀₁₆₀
maximal quotient of	Dic₁₀₀₈₀, D₂₀₁₆₀, D₃₀₂₄₀

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