#72220065

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₃×3₋¹⁺² ⋊ #120

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×AGL(1,ℤ/9)
commutator, cosocle	C₂²×C₃×C₉	C₂³×C₃
Frattini	C₂²×C₃	C₂×C₃×D₃²
Fitting	C₂³×C₃×3₋¹⁺²	C₂²
socle	C₂²×C₃²	C₂²×C₃×D₃
hypercenter	C₂³	D₃×AGL(1,ℤ/9)
hypocenter	C₃×C₉	C₃×#120
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂³

1195 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1195 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^2,3^3|a^2=f,g^3=i,b^a=be,c^a=cf,g^a=g^2i^2,g^b=g^2i^2,g^d=gi^2,h^a=hi^2,h^b=h^2,i^a=i^2,i^b=i^2

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220067, #72220105, #72220133, …
maximal normal subgroup of	#72220105, #72220133, #72220145
maximal quotient of	#72220067

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