#72231227

order	14400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₅² ⋊ C₂²×M₁₆
Sylow	C₅² ⋊ C₂×#63818
Sylow	C₃² ⋊ C₂×#11717347

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅×ΓL(1,𝔽₁₆)⋊D₃
commutator, socle	C₂×C₃²×C₅²	C₂³×C₄
Fitting	C₂²×C₃²×C₅²	C₂²×C₄
prime radical	C₂²×C₃⋊D₃×C₅⋊D₅	C₄
cosocle	C₃²×C₅×Dic₅	C₂⁴
hypercenter	C₂²	D₅×ΓL(1,𝔽₁₆)⋊D₃
hypocenter	C₃²×C₅²	C₂²×M₁₆
LCS	G ▷ C₂×C₃²×C₅² ▷ C₃²×C₅²
UCS	C₁ ◁ C₂ ◁ C₂²

3258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Polycyclic presentation:

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

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231278
maximal normal subgroup of	#72231278

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