#72254757

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#41×3₋¹⁺² ⋊ C₂
Sylow	C₃×3₋¹⁺² ⋊ AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	C₂×#10619980
commutator, cosocle	C₃×C₄×C₉	C₂³×C₃
Frattini	C₃×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃×#41×3₋¹⁺²	C₂
socle	C₂×C₃²	C₂×C₃×#67
hypercenter	#41	C₃⋊AGL(1,ℤ/9)
hypocenter	C₃×C₉	C₃×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 216

Polycyclic presentation:

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

Conjugacy classes & character table

183 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×C₃⋊AGL(1,ℤ/9)
maximal normal subgroup of	AGL(1,ℤ/8)×C₃⋊AGL(1,ℤ/9)
maximal quotient of	#72254760

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