Finite Group Databasebeta

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3−1+8 #72290073

order19683
nilpotency class2
automorphism?
homologyAbelianization (H1) = C38 · Schur multiplier (H2) = C327

Aliases

extraspecial3−1+8

Subgroups

subgroupG/H
center, commutator, Frattini, socle, cosocleC3C38
prime radicalC3×H(7,𝔽3)C3
LCSG ▷ C3 ▷ C1
UCSC1 ◁ C3 ◁ G
LECSG ▷ C3 ▷ C1

127972082 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 243

Polycyclic presentation:

3^9|a^3=i,b^a=bi,d^c=di,f^e=fi,h^g=hi

Conjugacy classes & character table

6563 conjugacy classes. The character table is too large to display.

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