D₆₈₄₀ #72212426

order	13680
automorphism	Aut(G) order 11819520
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₆₈₄₀

Splittings

Fitting	C₅×C₈×C₉×C₁₉ ⋊ C₂
Hall	C₅×C₉×C₁₉ ⋊ D₈
Hall	C₉×C₁₉ ⋊ D₄₀
Hall	C₅×C₁₉ ⋊ D₇₂
Hall	C₅×C₉ ⋊ D₁₅₂
Sylow	C₁₉ ⋊ D₃₆₀
Sylow	C₉ ⋊ D₇₆₀
Sylow	C₅ ⋊ D₁₃₆₈

Subgroups

	subgroup	G/H
center	C₂	D₃₄₂₀
commutator, cosocle	C₄×C₅×C₉×C₁₉	C₂²
Frattini	C₃×C₄	C₂×D₂₈₅
Fitting	C₅×C₈×C₉×C₁₉	C₂
socle	C₂×C₃×C₅×C₁₉	D₁₂
hypercenter	C₈	D₈₅₅
hypocenter	C₅×C₉×C₁₉	D₈
LCS	G ▷ C₄×C₅×C₉×C₁₉ ▷ C₂×C₅×C₉×C₁₉ ▷ C₅×C₉×C₁₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 6840

A minimal faithful permutation representation — 2 generators on 41 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);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,22)(20,21)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)

Polycyclic presentation:

2^4,3,5,19,3|b^2=cd,c^2=d,e^3=h,b^a=bc,c^a=cd,e^a=e^2h^2,f^a=f^4,g^a=g^18,h^a=h^2

Conjugacy classes & character table

3423 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₃₆₈₀, D₂₀₅₂₀
maximal normal subgroup of	D₁₃₆₈₀
maximal quotient of	D₁₃₆₈₀, Dic₆₈₄₀, D₂₀₅₂₀

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, 3, 5, 19, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.8);;
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*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^18);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D6840 #72212426