#72231626

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₃×C₄

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₅ ⋊ #384
Sylow	C₃² ⋊ #65688
Sylow	C₅ ⋊ #10592674

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×C₃⋊D₃×AGL(1,𝔽₅)
commutator, socle	C₂²×C₃²×C₅	C₂²×C₄
Fitting	C₂×C₃²×C₄×C₅	C₂×C₄
prime radical, cosocle	C₂²×C₃²×D₅	C₂³
hypercenter	C₂×C₄	C₃⋊D₃×AGL(1,𝔽₅)
hypocenter	C₃²×C₅	#384
LCS	G ▷ C₂²×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄

1116 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 270 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 720

Polycyclic presentation:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊D₃×#65688, AGL(1,𝔽₅)×#64149, #72231689
maximal normal subgroup of	C₃⋊D₃×#65688, AGL(1,𝔽₅)×#64149, #72231689

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