#72220084

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

#6900 ⋊ #99

Subgroups

	subgroup	G/H
center	C₂×C₃	C₃×#3676⋊C₃
commutator	C₂²×C₃×C₉	C₂×C₃²×C₄
Frattini	C₂³×C₃²	C₂×C₃²×D₃
Fitting	C₂×D₄×#6900	C₂
socle	C₂×C₃²	C₃²×#217
prime radical	C₂×C₃³×D₄	C₃×D₃
cosocle	C₂³×C₃×C₉	C₂²×C₃²
hypercenter	C₂×D₄×C₉	AGL(1,ℤ/9)
hypocenter	C₉	3₋¹⁺²×#99
LCS	G ▷ C₂²×C₃×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₉ ◁ C₂×D₄×C₉

1245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

321 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220126, #72220137, #72220158
maximal normal subgroup of	#72220126, #72220137, #72220158

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, 1, F.6);;
gap> SetPower(coll, 4, F.7^2);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
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, 2, 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);