D₂₆₂₈₀ #72289640

order	52560
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₂₈₀

Splittings

Fitting	C₅×C₈×C₉×C₇₃ ⋊ C₂
Hall	C₅×C₉×C₇₃ ⋊ D₈
Hall	C₉×C₇₃ ⋊ D₄₀
Hall	C₅×C₇₃ ⋊ D₇₂
Sylow	C₇₃ ⋊ D₃₆₀
Hall	C₅×C₉ ⋊ D₅₈₄
Sylow	C₉ ⋊ D₂₉₂₀
Sylow	C₅ ⋊ D₅₂₅₆

Subgroups

	subgroup	G/H
center	C₂	D₁₃₁₄₀
commutator, cosocle	C₄×C₅×C₉×C₇₃	C₂²
Frattini	C₃×C₄	C₂×D₁₀₉₅
Fitting	C₅×C₈×C₉×C₇₃	C₂
socle	C₂×C₃×C₅×C₇₃	D₁₂
hypercenter	C₈	D₃₂₈₅
hypocenter	C₅×C₉×C₇₃	D₈
LCS	G ▷ C₄×C₅×C₉×C₇₃ ▷ C₂×C₅×C₉×C₇₃ ▷ C₅×C₉×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 26280

A minimal faithful permutation representation — 2 generators on 95 points:

(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17)(18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,22)(20,21)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2^4,3,5,73,3|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^72,h^a=h^2

Conjugacy classes & character table

13143 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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 73, 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.7^72);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26280 #72289640