#72222927

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃×(C₃⋊D₃)⋊A₄
commutator, Fitting, cosocle	C₂³×C₃³	C₂²×C₃
Frattini	C₂×C₃	D₃²×A₄
socle	C₂³×C₃²	C₂×C₃×D₃
hypocenter	C₂²×C₃³	C₃×D₄
LCS	G ▷ C₂³×C₃³ ▷ C₂²×C₃³
UCS	C₁ ◁ C₂

918 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 52 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:

2^2,3,2^3,3^3|b^a=bd,e^c=ef,f^c=e,g^b=g^2,g^c=gi,h^a=h^2,h^b=h^2,i^b=i^2

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222930, #72222984, #72222986
maximal normal subgroup of	#72222930, #72222984, #72222986

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