Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C81 · Schur multiplier (H2) = C32 |
| direct product | C3×C243⋊C3 |
| unambiguous semidirect product | C243⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C81 | C32 |
| commutator | C3 | C32×C81 |
| Frattini, cosocle | C81 | C33 |
| socle | C32 | C3×C81 |
| prime radical | C33 | C81 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C81 ◁ G | |
| LECS | G ▷ C81 ▷ C27 ▷ C9 ▷ C3 ▷ C1 | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 246 |
| transitive degree | μt(G) = 729 |
891 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 5842);