#72230601

order	4800
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅² ⋊ C₄×#41
Sylow	C₅² ⋊ C₄×#237
Sylow	C₃ ⋊ #13107559

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂×C₃×C₅²	C₂³×C₄
Fitting	C₃×C₄×C₅²	C₂²×C₄
prime radical	#72230591	C₂
cosocle	C₂×C₃×C₅⋊D₅	C₂⁴
hypercenter	C₄	D₃×D₅⋊AGL(1,𝔽₅)
hypocenter	C₃×C₅²	C₄×#41
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂ ◁ C₄

1456 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^2=e,d^2=f,b^a=bf,d^a=df,g^a=g^2,g^b=g^2,h^a=h^4,h^c=h^3,h^e=h^4,i^a=h^2i,i^c=i^3,i^e=i^4

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#13107559, #72230980, #72259698, …
maximal normal subgroup of	D₃×#13107559, #72230980, #72259698, …

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