#72220135

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₃×3₋¹⁺² ⋊ B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#72220033
commutator, cosocle	C₂³×C₃×C₉	C₂³×C₃
Frattini	C₂³×C₃	C₂×C₃×D₃²
Fitting	C₂×C₃×D₄×3₋¹⁺²	C₂²
socle	C₂×C₃²	C₃×#953
hypercenter	C₂×D₄	D₃×AGL(1,ℤ/9)
hypocenter	C₃×C₉	C₃×B(4,𝔽₂)
LCS	G ▷ C₂³×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×D₄

1746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^3,3^3|a^2=g,h^3=j,c^a=cf,c^b=ce,e^a=eg,f^b=fg,h^c=h^2j^2,h^d=hj^2,i^a=i^2,i^c=i^2,j^c=j^2

Conjugacy classes & character table

189 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220137, #72220179
maximal normal subgroup of	#72220179
maximal quotient of	#72220137

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