#72231116

order	4800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₅²×D₄ ⋊ C₂²
Hall	C₃×C₅² ⋊ C₂×#127
Sylow	C₅² ⋊ D₃×#127
Sylow	C₃ ⋊ #127×C₅⋊D₅

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₃×C₅²	C₂⁴
Fitting	C₂×C₃×C₅²×D₄	C₂²
hypercenter	C₂×D₄	D₃×C₅⋊D₅
hypocenter	C₃×C₅²	C₂×#127
LCS	G ▷ C₂²×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂² ◁ C₂×D₄

3984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 1200

Polycyclic presentation:

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

Conjugacy classes & character table

330 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#127×C₅⋊D₅
maximal normal subgroup of	D₃×#127×C₅⋊D₅

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