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H(3,𝔽27) #72237348

order19683
nilpotency class2
automorphism?
homologyAbelianization (H1) = C36 · Schur multiplier (H2) = C318

Aliases

unitriangularUT(3,𝔽27)
HeisenbergH(3,𝔽27)

Subgroups

subgroupG/H
center, commutator, Frattini, socle, cosocleC33C36
LCSG ▷ C33 ▷ C1
UCSC1 ◁ C33 ◁ G
LECSG ▷ C33 ▷ C1

193278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 729

Polycyclic presentation:

3^9|d^a=dg,d^b=dgh^2,d^c=di,e^a=egh^2,e^b=ei,e^c=eh^2,f^a=fi,f^b=fh^2,f^c=fg^2hi

Conjugacy classes & character table

755 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, [3, 3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.7^2*F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);