#72220060

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₉⋊C₉ ⋊ C₂
Sylow	C₉⋊C₉ ⋊ #120

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂²×AGL(1,ℤ/9)
commutator	C₂²×C₉	C₂³×C₉
Frattini, socle	C₂²×C₃²	C₂²×C₃×D₃
Fitting	C₂⁴×C₉⋊C₉	C₂
prime radical	C₃×#63515	C₃
cosocle	C₂²×C₃×C₉	C₂³×C₃
hypercenter	C₂⁴×C₃	AGL(1,ℤ/9)
hypocenter	C₉	C₉×#120
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃

853 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 853 conjugacy classes of subgroups — too many to draw. Only its 153 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,d^3=g,h^3=i,b^a=be,c^a=cf,d^a=di^2,h^a=h^2i^2,h^d=hi^2,i^a=i^2

Conjugacy classes & character table

318 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220063, #72220121, #72229378, …
maximal normal subgroup of	#72220063, #72220121

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