#72259333

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

C₃²×C₉² ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃²×C₉²	C₂²
Frattini	C₃²	C₃⋊#1113
socle	C₃⁴	D₃²
LCS	G ▷ C₃²×C₉²
UCS	C₁

793 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 793 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 108

Polycyclic presentation:

2^2,3^6|c^3=g,d^3=h,c^a=c^2g^2,c^b=c^2g^2,d^a=d^2h^2,d^b=c^2d,e^a=e^2,e^b=e^2,f^a=e^2f,f^b=f^2,g^a=g^2,g^b=g^2,h^a=h^2,h^b=g^2h

Conjugacy classes & character table

213 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259335, D₃×D₉×C₃⋊D₉, D₃×#11757745, …
maximal normal subgroup of	#72259335, D₃×D₉×C₃⋊D₉, D₃×#11757745, …

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