D₃₂₆₈₀ #72304191

order	65360
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₆₈₀

Splittings

Fitting	C₅×C₈×C₁₉×C₄₃ ⋊ C₂
Hall	C₅×C₁₉×C₄₃ ⋊ D₈
Hall	C₁₉×C₄₃ ⋊ D₄₀
Hall	C₅×C₄₃ ⋊ D₁₅₂
Hall	C₅×C₁₉ ⋊ D₃₄₄
Sylow	C₄₃ ⋊ D₇₆₀
Sylow	C₁₉ ⋊ D₁₇₂₀
Sylow	C₅ ⋊ D₆₅₃₆

Subgroups

	subgroup	G/H
center	C₂	D₁₆₃₄₀
commutator, cosocle	C₄×C₅×C₁₉×C₄₃	C₂²
Frattini	C₄	C₂×D₄₀₈₅
Fitting	C₅×C₈×C₁₉×C₄₃	C₂
socle	C₂×C₅×C₁₉×C₄₃	D₄
hypercenter	C₈	D₄₀₈₅
hypocenter	C₅×C₁₉×C₄₃	D₈
LCS	G ▷ C₄×C₅×C₁₉×C₄₃ ▷ C₂×C₅×C₁₉×C₄₃ ▷ C₅×C₁₉×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 32680

A minimal faithful permutation representation — 2 generators on 75 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(2,8)(3,7)(4,6)(10,13)(11,12)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2^4,5,19,43|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^18,g^a=g^42

Conjugacy classes & character table

16343 conjugacy classes. The character table is too large to display.

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 5, 19, 43]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32680 #72304191