Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C81 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | 3−1+2⋊C81 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C27 | H(3,𝔽3) |
| commutator | C32 | C3×C81 |
| Frattini, cosocle | C32×C27 | C32 |
| socle | C32 | C32⋊C27 |
| prime radical | C33 | C3×C27 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C27 ◁ C32×C27 ◁ G | |
| LECS | G ▷ C32×C27 ▷ C3×C9 ▷ C3 ▷ C1 | |
85 conjugacy classes of subgroups.
| transitive degree | μt(G) = 729 |
Polycyclic presentation:
3^7|a^3=d,b^3=e,d^3=f,f^3=g,b^a=bc,c^a=ce,c^b=ce459 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 120);