Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C92 · Schur multiplier (H2) = C35 |
| direct product | C3×#10624466 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C34 | H(3,𝔽3) |
| commutator | C32 | C3×C92 |
| Frattini, cosocle | C34 | C33 |
| prime radical | C35 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C34 ◁ C35 ◁ G | |
| LECS | G ▷ C34 ▷ C3 ▷ C1 | |
2384 conjugacy classes of subgroups; only the 494 normal subgroups are shown.
This group has 2384 conjugacy classes of subgroups — too many to draw. Only its 494 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 729 |
459 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 4078);