#72220022

order	15552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	#127×#6900 ⋊ C₂
Sylow	#6900 ⋊ #453

Subgroups

	subgroup	G/H
center	C₂×C₃	C₃×#63499⋊C₃
commutator, cosocle	C₂×C₃×C₄×C₉	C₂³×C₃²
Frattini	C₂×C₃²×C₄	C₂²×C₃²×D₃
Fitting	#127×#6900	C₂
socle	C₂×C₃²	C₃²×#937
prime radical	C₃×#10588757⋊C₃	C₃
hypercenter	C₉×#127	AGL(1,ℤ/9)
hypocenter	C₉	3₋¹⁺²×#453
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₄×C₉ ◁ C₉×#127

3258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3^2,2^2,3,2,3^2|b^2=f,e^3=h^2,g^2=i,j^3=k,b^a=bg,c^b=cf,e^a=ek^2,e^d=eh,f^a=fi,g^a=gi,g^c=gi,j^a=j^2k^2,j^d=jk^2,k^a=k^2

Conjugacy classes & character table

456 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220092
maximal normal subgroup of	#72220092

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 3]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 5, F.8^2);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.11^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2*F.11^2);;
gap> SetConjugate(coll, 10, 4, F.10*F.11^2);;
gap> SetConjugate(coll, 11, 1, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);