#72259785

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃²

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃²×H(3,𝔽₃) ⋊ C₂²
Sylow	C₃²×H(3,𝔽₃) ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₃	C₂×D₃×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃²×H(3,𝔽₃)	C₂³
Fitting	C₂²×C₃²×H(3,𝔽₃)	C₂²
socle	C₂×C₃³	C₂²×C₃⋊D₃
hypercenter	C₂²×C₃	D₃×C₃⋊(C₃⋊D₃)
hypocenter	C₃²×H(3,𝔽₃)	C₂×D₄
LCS	G ▷ C₂×C₃²×H(3,𝔽₃) ▷ C₃²×H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃

4530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 72

Polycyclic presentation:

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72259880

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