#72260182

order	5760
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×C₅×D₄ ⋊ C₂²×C₄
Hall	C₃²×C₅ ⋊ C₂×#518
Sylow	C₃² ⋊ C₂×#66258
Sylow	C₅ ⋊ #11900008

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×D₃²×AGL(1,𝔽₅)
commutator, socle	C₂×C₃²×C₅	C₂⁴×C₄
Fitting	C₃²×C₅×D₄	C₂²×C₄
prime radical	D₃²×D₄×D₅	C₂
cosocle	C₂×C₃²×D₅	C₂⁵
hypercenter	D₄	D₃²×AGL(1,𝔽₅)
hypocenter	C₃²×C₅	C₂×#518
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ D₄

8106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Polycyclic presentation:

2^7,3^2,5|a^2=g,d^2=f,c^a=cg,d^a=dg,d^b=dg,h^b=h^2,h^e=h^2,i^b=h^2i,i^e=i^2,j^d=j^3,j^f=j^4

Conjugacy classes & character table

201 conjugacy classes — view the full character table.

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", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 3, 5]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> SetConjugate(coll, 9, 2, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> SetConjugate(coll, 10, 4, F.10^3);;
gap> SetConjugate(coll, 10, 6, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);