#72220064

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

Aliases

No aliases for this group.

Splittings

Sylow

C₉² ⋊ #120

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₉²
commutator, cosocle	C₂²×C₉²	C₂³
Frattini, socle	C₂²×C₃²	C₂×D₃²
Fitting	C₂³×C₉²	C₂²
hypercenter	C₂³	D₉²
hypocenter	C₉²	#120
LCS	G ▷ C₂²×C₉² ▷ C₉²
UCS	C₁ ◁ C₂² ◁ C₂³

1054 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1054 conjugacy classes of subgroups — too many to draw. Only its 102 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:

2^5,3^4|a^2=e,f^3=h,g^3=i,b^a=bd,c^a=ce,f^a=f^2h^2,f^b=f^2h^2,g^a=f^2g,g^b=g^2i^2,h^a=h^2,h^b=h^2,i^a=h^2i,i^b=i^2

Conjugacy classes & character table

222 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220067, #72220102, #72220131, …
maximal normal subgroup of	#72220067, #72220102, #72220131, …

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