#72247228

order	4860
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃²

Aliases

No aliases for this group.

Splittings

Fitting, Hall	C₅×3₋¹⁺⁴ ⋊ C₄
derived	C₃×C₅×C₉ ⋊ C₃²×C₄
Sylow	3₋¹⁺⁴ ⋊ AGL(1,𝔽₅)
Sylow	C₅ ⋊ #66409⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₃×C₅×C₉	C₃²×C₄
Frattini	C₃	C₃²×C₃⋊ΓL(1,𝔽₁₆)
Fitting	C₅×3₋¹⁺⁴	C₄
socle	C₃×C₅	C₃²×C₃⋊Dic₃
prime radical	C₃×D₅×H(3,𝔽₃)	D₃
cosocle	C₃×C₉×D₅	C₂×C₃²
LCS	G ▷ C₃×C₅×C₉
UCS	C₁

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 135

Polycyclic presentation:

2,3^2,2,3^2,5,3|a^2=d,f^3=h,c^a=ch^2,e^a=e^2,e^b=eh^2,f^a=f^2h^2,f^b=fh,f^c=fh,g^a=g^3,g^d=g^4,h^a=h^2

Conjugacy classes & character table

193 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#4207⋊C₃, #72247336, #72254467, …
maximal normal subgroup of	AGL(1,𝔽₅)×#4207⋊C₃, #72254467, #72254469

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