#72220063

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×#6900 ⋊ C₂
Sylow	#6900 ⋊ #120

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂²×C₃×AGL(1,ℤ/9)
commutator, cosocle	C₂²×C₃×C₉	C₂³×C₃²
Frattini, socle	C₂²×C₃²	C₂²×C₃²×D₃
Fitting	C₂⁴×#6900	C₂
prime radical	C₃×#11721704	C₃
hypercenter	C₂⁴×C₉	AGL(1,ℤ/9)
hypocenter	C₉	3₋¹⁺²×#120
LCS	G ▷ C₂²×C₃×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₉

3101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3^2,2^2,3^3|a^2=g,e^3=h^2,i^3=j,b^a=bf,c^a=cg,e^a=ej^2,e^d=eh,i^a=i^2j^2,i^d=ij^2,j^a=j^2

Conjugacy classes & character table

474 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220108, #72220126, #72220142
maximal normal subgroup of	#72220108, #72220126, #72220142

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