D₂₅₅₆₀ #72288710

order	51120
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₅₅₆₀

Splittings

Fitting	C₅×C₈×C₉×C₇₁ ⋊ C₂
Hall	C₅×C₉×C₇₁ ⋊ D₈
Hall	C₉×C₇₁ ⋊ D₄₀
Hall	C₅×C₇₁ ⋊ D₇₂
Sylow	C₇₁ ⋊ D₃₆₀
Hall	C₅×C₉ ⋊ D₅₆₈
Sylow	C₉ ⋊ D₂₈₄₀
Sylow	C₅ ⋊ D₅₁₁₂

Subgroups

	subgroup	G/H
center	C₂	D₁₂₇₈₀
commutator, cosocle	C₄×C₅×C₉×C₇₁	C₂²
Frattini	C₃×C₄	C₂×D₁₀₆₅
Fitting	C₅×C₈×C₉×C₇₁	C₂
socle	C₂×C₃×C₅×C₇₁	D₁₂
hypercenter	C₈	D₃₁₉₅
hypocenter	C₅×C₉×C₇₁	D₈
LCS	G ▷ C₄×C₅×C₉×C₇₁ ▷ C₂×C₅×C₉×C₇₁ ▷ C₅×C₉×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 25560

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

Polycyclic presentation:

2^4,3,5,71,3|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^70,h^a=h^2

Conjugacy classes & character table

12783 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, 71, 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.7^70);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D25560 #72288710