#72219959

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₄²×C₉⋊C₉ ⋊ C₂
Sylow	C₉⋊C₉ ⋊ #104

Subgroups

	subgroup	G/H
center	C₃×C₄	C₄⋊AGL(1,ℤ/9)
commutator	C₄×C₉	C₂×C₄×C₉
Frattini	C₂×C₃²×C₄	C₂×C₃×D₃
Fitting	C₄²×C₉⋊C₉	C₂
socle	C₂×C₃²	C₃×(C₂×D₃)⋊C₄
prime radical	C₃×#3697	C₂×C₃
cosocle	C₂×C₃×C₄×C₉	C₂²×C₃
hypercenter	C₃×C₄²	AGL(1,ℤ/9)
hypocenter	C₉	C₉×#104
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₃×C₄²

258 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 71 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:

2^2,3,2^2,3,2,3^2|b^2=d,c^3=f,e^2=g,h^3=i,b^a=be,c^a=ci^2,d^a=dg,e^a=eg,h^a=h^2i^2,h^c=hi^2,i^a=i^2

Conjugacy classes & character table

318 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219963, #72220017
maximal normal subgroup of	#72219963, #72220017

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