#72247336

order	14580
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, derived, Hall	C₅×#10624749 ⋊ C₄
Sylow	#10624749 ⋊ AGL(1,𝔽₅)
Sylow	C₅ ⋊ #72228525

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, hypocenter	C₅×#10624749	C₄
Frattini	C₃³	C₃⋊(C₃⋊ΓL(1,𝔽₁₆))
socle	C₃×C₅	#11757642
prime radical, cosocle	D₅×#10624749	C₂
LCS	G ▷ C₅×#10624749
UCS	C₁

1096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 135

Polycyclic presentation:

2^2,3^3,5,3^3|a^2=b,e^3=i,c^a=c^2,d^a=d^2,d^c=dh,e^a=e^2i^2,e^d=eg,f^a=f^3,f^b=f^4,g^c=gi,h^e=hi^2,i^a=i^2

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#13091120
maximal normal subgroup of	AGL(1,𝔽₅)×#13091120

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