Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #6878 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27 · Schur multiplier (H2) = C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C27 | #732 |
| commutator | C3×C9 | C3×C27 |
| Frattini, cosocle | C32×C27 | C32 |
| socle | C3 | H(3,𝔽3)⋊C27 |
| prime radical | #6908 | C9 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ C3×C27 ◁ C32×C27 ◁ G | |
| LECS | G ▷ C32×C27 ▷ C3×C9 ▷ C3 ▷ C1 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 243 |
| transitive degree | μt(G) = 243 |
Polycyclic presentation:
3^7|a^3=d,c^3=g^2,d^3=f,f^3=g,b^a=bc,c^a=ce,e^a=eg,c^b=ce,e^b=eg315 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 118);