#72259996

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · 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	C₂	C₂×D₃×C₃⋊((C₃⋊D₃)⋊C₃)
commutator, cosocle	C₂×C₃⁴	C₂³×C₃
Frattini	C₂×C₃	C₂×C₃×D₃×C₃⋊D₃
Fitting	C₂²×C₃²×H(3,𝔽₃)	C₂²
socle	C₂×C₃³	C₂²×C₃×D₃
hypercenter	C₂²	D₃×C₃⋊((C₃⋊D₃)⋊C₃)
hypocenter	C₃⁴	C₂×C₃×D₄
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

2810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 108

Polycyclic presentation:

2^3,3,2,3^4|b^a=be,f^c=f^2,f^d=fi,g^c=g^2,h^a=h^2,h^c=h^2,i^c=i^2

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

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