#72252808

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	3₋¹⁺²×#493 ⋊ C₂
Sylow	3₋¹⁺² ⋊ #3184

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×AGL(1,ℤ/9)
commutator, cosocle	C₂×C₄×C₉	C₂⁴×C₃
Frattini	C₂×C₃×C₄	C₂³×C₃×D₃
Fitting	3₋¹⁺²×#493	C₂
socle	C₂²×C₃	C₂×C₃×D₃×D₄
hypercenter	#493	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#3184
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

1419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 576

Polycyclic presentation:

2^4,3,2^3,3^2|b^2=g,c^2=fh,d^2=g,f^2=h,i^3=j,c^a=cf,c^b=ch,d^a=dg,f^a=fh,i^b=i^2j^2,i^e=ij,j^b=j^2

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/9)×#3184, #72268944, #72268948
maximal normal subgroup of	AGL(1,ℤ/9)×#3184, #72268944, #72268948

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