#72219911

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₂×D₄×#6900 ⋊ C₂
Sylow	#6900 ⋊ AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×C₃×#611⋊C₃
commutator, cosocle	C₃×C₄×C₉	C₂³×C₃²
Frattini	C₃²×C₄	C₂²×C₃²×D₃
Fitting	C₂×D₄×#6900	C₂
socle	C₂×C₃²	C₂×C₃²×#67
prime radical	C₃×#63495⋊C₃	C₃
hypercenter	C₂×D₄×C₉	AGL(1,ℤ/9)
hypocenter	C₉	3₋¹⁺²×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₄×C₉ ◁ C₂×D₄×C₉

1437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3^2,2,3,2,3^2|a^2=f,e^3=g^2,f^2=h,i^3=j,b^a=bfh,c^a=ch,e^a=ej^2,e^d=eg,f^b=fh,i^a=i^2j^2,i^d=ij^2,j^a=j^2

Conjugacy classes & character table

321 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219937, #72220022, #72220045
maximal normal subgroup of	#72219937, #72220022, #72220045

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