#72254713

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₈×3₋¹⁺² ⋊ C₂
Sylow	C₃×3₋¹⁺² ⋊ #102

Subgroups

	subgroup	G/H
center	C₂²	C₄⋊(C₃⋊AGL(1,ℤ/9))
commutator	C₃×C₄×C₉	C₂×C₃×C₄
Frattini	C₂×C₃×C₄	C₂×C₃×C₃⋊D₃
Fitting	C₂×C₃×C₈×3₋¹⁺²	C₂
socle	C₂²×C₃²	C₃×D₁₂
prime radical	C₂×C₄⋊(C₃⋊AGL(1,ℤ/9))	C₂
cosocle	C₂×C₃×C₄×C₉	C₂²×C₃
hypercenter	C₂×C₈	C₃⋊AGL(1,ℤ/9)
hypocenter	C₃×C₉	C₃×#102
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 432

Polycyclic presentation:

2^2,3,2^3,3^3|a^2=d,b^2=def,e^2=f,g^3=i,b^a=be,e^a=ef,g^a=g^2i^2,g^c=gi,h^a=h^2,i^a=i^2

Conjugacy classes & character table

282 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254723, #102×C₃⋊AGL(1,ℤ/9), #72254769
maximal normal subgroup of	#72254723, #102×C₃⋊AGL(1,ℤ/9), #72254769

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