#72260199

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₅ ⋊ C₂×#104
Sylow	C₃² ⋊ D₅⋊#104
Sylow	C₅ ⋊ C₂×#63568

Subgroups

	subgroup	G/H
center	C₄	(C₂×D₃)⋊ΓL(1,𝔽₁₆)
commutator	C₃²×C₄×C₅	C₂²×C₄
Frattini	C₄	C₂×D₃×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃²×C₄×C₅	C₂×C₄
socle	C₂×C₃²×C₅	C₂×C₂²⋊C₄
prime radical	C₃×D₅×#235	C₂
cosocle	C₃²×C₄×D₅	C₂³
hypercenter	C₂×C₄	D₃×ΓL(1,𝔽₁₆)
hypocenter	C₃²×C₅	C₂×#104
LCS	G ▷ C₃²×C₄×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄

788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 120

Polycyclic presentation:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260231, #72260271, AGL(1,𝔽₅)×#63568
maximal normal subgroup of	#72260231, #72260271, AGL(1,𝔽₅)×#63568

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.6);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 2, 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);