#72252904

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Splittings

Fitting	3₋¹⁺²×#322 ⋊ C₂
Sylow	3₋¹⁺² ⋊ #1455

Subgroups

	subgroup	G/H
center	C₄²	C₂²×AGL(1,ℤ/9)
commutator	C₂²×C₉	C₂×C₃×C₄²
Frattini	C₃×C₄²	C₂²×C₃×D₃
Fitting	3₋¹⁺²×#322	C₂
socle	C₂²×C₃	C₃×C₄²×D₃
prime radical	C₂²×AGL(1,ℤ/9)	C₄²
cosocle	C₄²×C₉	C₂³×C₃
hypercenter	#322	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#1455
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₄² ◁ #322

675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 576

Polycyclic presentation:

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

Conjugacy classes & character table

328 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/9)×#1455, #72268944, #72268961
maximal normal subgroup of	AGL(1,ℤ/9)×#1455, #72268944, #72268961

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