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) = C33×C9 |
| direct product | C9×H(3,𝔽3)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C9 | H(3,𝔽3) |
| commutator | C32 | C3×C92 |
| Frattini, cosocle | C34 | C33 |
| socle | C33 | C3×H(3,𝔽3) |
| prime radical | C32×H(3,𝔽3) | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C9 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C34 ▷ C3 ▷ C1 | |
815 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 815 conjugacy classes of subgroups — too many to draw. Only its 206 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9);(10,11,13,12,14,16,15,17,18)(19,20,23,21,25,24,22,26,27);(19,20,24)(21,25,27)(22,26,23)459 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 4114);