D₁₀₂₀ #72210102

order	2040
automorphism	Aut(G) order 261120
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₂₀

Splittings

Fitting	C₃×C₄×C₅×C₁₇ ⋊ C₂
Hall	C₃×C₅×C₁₇ ⋊ D₄
Hall	C₅×C₁₇ ⋊ D₁₂
Hall	C₃×C₁₇ ⋊ D₂₀
Sylow	C₁₇ ⋊ D₆₀
Hall	C₃×C₅ ⋊ D₆₈
Sylow	C₅ ⋊ D₂₀₄
Sylow	C₃ ⋊ D₃₄₀

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₅₅
commutator, socle, cosocle	C₂×C₃×C₅×C₁₇	C₂²
Fitting	C₃×C₄×C₅×C₁₇	C₂
hypercenter	C₄	D₂₅₅
hypocenter	C₃×C₅×C₁₇	D₄
LCS	G ▷ C₂×C₃×C₅×C₁₇ ▷ C₃×C₅×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 1020

A minimal faithful permutation representation — 2 generators on 29 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);(2,4)(6,7)(9,12)(10,11)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)

Polycyclic presentation:

2^3,3,5,17|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^16

Conjugacy classes & character table

513 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃₀₆₀, D₂₀₄₀, D₅₁₀₀, …
maximal normal subgroup of	D₂₀₄₀
maximal quotient of	D₃₀₆₀, D₂₀₄₀, D₅₁₀₀, …

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, 3, 5, 17]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1020 #72210102