#72219963

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₄²×#6900 ⋊ C₂
Sylow	#6900 ⋊ #104

Subgroups

	subgroup	G/H
center	C₃×C₄	C₃×C₄⋊AGL(1,ℤ/9)
commutator	C₃×C₄×C₉	C₂×C₃²×C₄
Frattini	C₂×C₃²×C₄	C₂×C₃²×D₃
Fitting	C₄²×#6900	C₂
socle	C₂×C₃²	C₃²×(C₂×D₃)⋊C₄
prime radical	C₃×#3697⋊C₃	C₂×C₃
cosocle	C₂×C₃×C₄×C₉	C₂²×C₃²
hypercenter	C₄²×C₉	AGL(1,ℤ/9)
hypocenter	C₉	3₋¹⁺²×#104
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₄×C₉ ◁ C₄²×C₉

873 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3^2,2^2,3,2,3^2|b^2=e,d^3=g^2,f^2=h,i^3=j,b^a=bf,d^a=dj^2,d^c=dg,e^a=eh,f^a=fh,i^a=i^2j^2,i^c=ij^2,j^a=j^2

Conjugacy classes & character table

474 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220005, #72220022, #72220028
maximal normal subgroup of	#72220005, #72220022, #72220028

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