#72219975

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

Aliases

No aliases for this group.

Splittings

Sylow

C₉⋊C₉ ⋊ #100

Subgroups

	subgroup	G/H
center	C₂×C₃	#3676⋊C₃
commutator	C₂²×C₉	C₂×C₄×C₉
Frattini	C₂×C₃²×C₄	C₂×C₃×D₃
Fitting	C₂×D₄×C₉⋊C₉	C₂
socle	C₂×C₃²	C₃×#217
prime radical	C₂×C₃²×D₄	C₃×D₃
cosocle	C₂×C₃×C₄×C₉	C₂²×C₃
hypercenter	C₂×C₃×D₄	AGL(1,ℤ/9)
hypocenter	C₉	C₉×#100
LCS	G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×D₄

315 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 81 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|a^2=d,c^3=f,d^2=g,e^2=g,h^3=i,b^a=bdeg,c^a=ci^2,d^b=dg,e^a=eg,e^b=eg,h^a=h^2i^2,h^c=hi^2,i^a=i^2

Conjugacy classes & character table

219 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219979, #72220017
maximal normal subgroup of	#72219979, #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, 1, F.4);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.9^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, 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);