#72259553

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃²×3₋¹⁺² ⋊ C₂²
Sylow	C₃²×3₋¹⁺² ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂×C₃⋊D₃×AGL(1,ℤ/9)
commutator, cosocle	C₂×C₃²×C₉	C₂³×C₃
Frattini	C₂×C₃	C₂×C₃×D₃×C₃⋊D₃
Fitting	C₂²×C₃²×3₋¹⁺²	C₂²
socle	C₂×C₃³	C₂²×C₃×D₃
hypercenter	C₂²	C₃⋊D₃×AGL(1,ℤ/9)
hypocenter	C₃²×C₉	C₂×C₃×D₄
LCS	G ▷ C₂×C₃²×C₉ ▷ C₃²×C₉
UCS	C₁ ◁ C₂ ◁ C₂²

1884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 108

Polycyclic presentation:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#88668⋊C₃, #72259878
maximal normal subgroup of	D₃×#88668⋊C₃
maximal quotient of	#72259878

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