#72254708

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₃×3₋¹⁺²×#106 ⋊ C₂
Sylow	C₃×3₋¹⁺² ⋊ #442

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₃⋊AGL(1,ℤ/9)
commutator	C₃×C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃×3₋¹⁺²×#106	C₂
socle	C₂²×C₃²	C₃×D₃×D₄
prime radical	C₂×#13047909	C₂
cosocle	C₂×C₃×C₄×C₉	C₂³×C₃
hypercenter	#106	C₃⋊AGL(1,ℤ/9)
hypocenter	C₃×C₉	C₃×#442
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

1482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 864

Polycyclic presentation:

2^3,3,2^3,3^3|a^2=e,b^2=e,c^2=f,f^2=g,h^3=j,c^a=cf,c^b=cg,f^a=fg,h^b=h^2j^2,h^d=hj,i^b=i^2,j^b=j^2

Conjugacy classes & character table

276 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#442×C₃⋊AGL(1,ℤ/9)
maximal normal subgroup of	#442×C₃⋊AGL(1,ℤ/9)

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 3, 3, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^2*F.10^2);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);