#72247334

order	4860
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, derived, Hall	C₅×C₃²⋊H(3,𝔽₃) ⋊ C₄
Sylow	C₃²⋊H(3,𝔽₃) ⋊ AGL(1,𝔽₅)
Sylow	C₅ ⋊ #11757642

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃²	C₃⋊(C₃⋊ΓL(1,𝔽₁₆))
commutator, Fitting, hypocenter	C₅×C₃²⋊H(3,𝔽₃)	C₄
socle	C₃²×C₅	C₃⋊(C₃⋊Dic₃)
prime radical, cosocle	D₅×C₃²⋊H(3,𝔽₃)	C₂
LCS	G ▷ C₅×C₃²⋊H(3,𝔽₃)
UCS	C₁ ◁ C₃²

838 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 135

Polycyclic presentation:

2^2,3^3,5,3^2|a^2=b,c^a=c^2,d^a=d^2,d^c=dh,e^a=e^2,e^d=eg,f^a=f^3,f^b=f^4

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#92338, #72254531, #72254535
maximal normal subgroup of	AGL(1,𝔽₅)×#92338, #72254531, #72254535
maximal quotient of	#72247336

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");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 5, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
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);;
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> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);