D₁₂₀₀ #72210133

order	2400
automorphism	Aut(G) order 384000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₂₀₀

Splittings

Fitting	C₃×C₁₆×C₂₅ ⋊ C₂
Hall	C₃×C₂₅ ⋊ D₁₆
Sylow	C₂₅ ⋊ D₄₈
Sylow	C₃ ⋊ D₄₀₀

Subgroups

	subgroup	G/H
center	C₂	D₆₀₀
commutator, cosocle	C₃×C₈×C₂₅	C₂²
Frattini	C₅×C₈	C₂×D₁₅
Fitting	C₃×C₁₆×C₂₅	C₂
socle	C₂×C₃×C₅	D₄₀
hypercenter	C₁₆	D₇₅
hypocenter	C₃×C₂₅	D₁₆
LCS	G ▷ C₃×C₈×C₂₅ ▷ C₃×C₄×C₂₅ ▷ C₂×C₃×C₂₅ ▷ C₃×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 1200

A minimal faithful permutation representation — 2 generators on 44 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

603 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₄₀₀, D₃₆₀₀, D₆₀₀₀, …
maximal normal subgroup of	D₂₄₀₀
maximal quotient of	D₂₄₀₀, D₃₆₀₀, Dic₁₂₀₀, …

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

D1200 #72210133