#72260195

order	4800
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₅²×D₄ ⋊ C₂³
Hall	C₃×C₅² ⋊ C₂×H(5,𝔽₂)
Sylow	C₅² ⋊ C₂×#1009
Sylow	C₃ ⋊ #13107821

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×D₃×D₅²
commutator, socle, cosocle	C₂×C₃×C₅²	C₂⁵
Fitting	C₃×C₅²×D₄	C₂³
hypercenter	D₄	D₃×D₅²
hypocenter	C₃×C₅²	C₂×H(5,𝔽₂)
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂ ◁ D₄

4064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Polycyclic presentation:

2^6,3,5^2|a^2=f,c^a=cf,d^a=df,d^b=df,g^c=g^2,h^b=h^4,h^e=h^4,i^b=h^2i,i^e=i^4

Conjugacy classes & character table

222 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260237
maximal normal subgroup of	#72260237

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