#72231684

order	5760
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₃×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×C₄²×C₅ ⋊ C₂×C₄
Hall	C₃²×C₅ ⋊ C₄×#127
Sylow	C₃² ⋊ AGL(1,𝔽₅)×#127
Sylow	C₅ ⋊ #11802516

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊D₃×AGL(1,𝔽₅)
commutator, socle	C₂²×C₃²×C₅	C₂³×C₄
Fitting	C₃²×C₄²×C₅	C₂×C₄
prime radical	D₅×#64083	C₂
cosocle	C₂²×C₃²×D₅	C₂⁴
hypercenter	C₄²	C₃⋊D₃×AGL(1,𝔽₅)
hypocenter	C₃²×C₅	C₄×#127
LCS	G ▷ C₂²×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂² ◁ C₄²

5544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 720

Polycyclic presentation:

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

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊D₃×AGL(1,𝔽₅)×#127
maximal normal subgroup of	C₃⋊D₃×AGL(1,𝔽₅)×#127

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