#72221986

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

C₃²⋊H(3,𝔽₃) ⋊ SD₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle	#66472	C₂²×C₃
Frattini, socle	C₃²	C₃×AΓL(1,𝔽₉)
Fitting	C₃²⋊H(3,𝔽₃)	SD₁₆
prime radical	#10620206⋊C₃	C₂
hypocenter	C₃⁴	C₃×SD₁₆
LCS	G ▷ #66472 ▷ C₃⋊(C₃⋊(C₃⋊D₃)) ▷ C₃⁴
UCS	C₁

155 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 155 conjugacy classes of subgroups — too many to draw. Only its 17 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:

2^2,3,2^2,3^4|b^2=e,d^2=e,b^a=bd,d^a=de,d^b=de,f^b=fg,f^c=fh,f^d=g^2,f^e=f^2,g^a=g^2,g^b=fg^2,g^c=ghi^2,g^d=f,g^e=g^2,h^b=h^2i^2,h^d=h^2i,h^e=h^2,i^a=h^2i^2,i^b=h^2i,i^d=hi,i^e=i^2

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72221987, #72222012
maximal normal subgroup of	#72222012

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