D₇₆₀₀ #72212845

order	15200
automorphism	Aut(G) order 21888000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₇₆₀₀

Splittings

Fitting	C₁₆×C₁₉×C₂₅ ⋊ C₂
Hall	C₁₉×C₂₅ ⋊ D₁₆
Sylow	C₂₅ ⋊ D₃₀₄
Sylow	C₁₉ ⋊ D₄₀₀

Subgroups

	subgroup	G/H
center	C₂	D₃₈₀₀
commutator, cosocle	C₈×C₁₉×C₂₅	C₂²
Frattini	C₅×C₈	C₂×D₉₅
Fitting	C₁₆×C₁₉×C₂₅	C₂
socle	C₂×C₅×C₁₉	D₄₀
hypercenter	C₁₆	D₄₇₅
hypocenter	C₁₉×C₂₅	D₁₆
LCS	G ▷ C₈×C₁₉×C₂₅ ▷ C₄×C₁₉×C₂₅ ▷ C₂×C₁₉×C₂₅ ▷ C₁₉×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 7600

A minimal faithful permutation representation — 2 generators on 60 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)(42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2^5,5,19,5|b^2=cd,c^2=de,d^2=e,f^5=h,b^a=bc,c^a=cd,d^a=de,f^a=f^4h^4,g^a=g^18,h^a=h^4

Conjugacy classes & character table

3803 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₅₂₀₀, D₂₂₈₀₀
maximal normal subgroup of	D₁₅₂₀₀
maximal quotient of	D₁₅₂₀₀, Dic₇₆₀₀, D₂₂₈₀₀

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, 5, 19, 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, 6, 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.6^4*F.8^4);;
gap> SetConjugate(coll, 7, 1, F.7^18);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D7600 #72212845