Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)2 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9 · Schur multiplier (H2) = C34 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C3×C9 | C34 |
| commutator, socle | C32 | C33×C9 |
| prime radical | C3×H(3,𝔽3) | C3×C9 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
542 conjugacy classes of subgroups; only the 410 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 410 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 729 |
Polycyclic presentation:
3^7|a^3=f,d^3=e,f^3=g,b^a=be,c^a=cg,d^a=de,c^b=ce,d^b=dg363 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 8105);