#72220185

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃×((C₂²⋊C₉)⋊C₂)⋊C₃
commutator, cosocle	C₂×C₃×C₂²⋊C₉	C₂²×C₃
Frattini	C₂×C₃	C₃×D₃×S₄
Fitting	C₂³×C₃³	C₂×D₃
socle	C₂³×C₃²	C₂×C₃×D₃
hypocenter	C₃×C₂²⋊C₉	C₃×D₄
LCS	G ▷ C₂×C₃×C₂²⋊C₉ ▷ C₃×C₂²⋊C₉
UCS	C₁ ◁ C₂

548 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 48 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:

2^2,3,2,3^3,2^2|e^3=g,b^a=bd,e^a=e^2g^2,e^b=e^2g^2,e^c=eg^2,f^a=fg^2,f^b=f^2,g^a=g^2,g^b=g^2,h^e=i,i^a=hi,i^b=hi,i^e=hi

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#67×((C₂²⋊C₉)⋊C₂)⋊C₃
maximal normal subgroup of	#67×((C₂²⋊C₉)⋊C₂)⋊C₃

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