Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9 · Schur multiplier (H2) = C37 |
| direct product | C32×H(3,𝔽3)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C34 | H(3,𝔽3) |
| commutator | C32 | C33×C9 |
| Frattini, cosocle | C33 | C34 |
| prime radical | C33×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C34 ◁ C35 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
4058 conjugacy classes of subgroups; only the 764 normal subgroups are shown.
This group has 4058 conjugacy classes of subgroups — too many to draw. Only its 764 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3);(4,5,6);(7,8,10,9,11,13,12,14,15)(16,17,20,18,22,21,19,23,24);(16,17,21)(18,22,24)(19,23,20)459 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 7979);