Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #14546264 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C34 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | #6866 |
| commutator | C34 | C3×C9 |
| Frattini, cosocle | C35 | C32 |
| prime radical | C3×C32⋊H(3,𝔽3) | C3 |
| LCS | G ▷ C34 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ C35 ◁ G | |
| LECS | G ▷ C35 ▷ C33 ▷ C3 ▷ C1 | |
749 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 749 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 81 |
Polycyclic presentation:
3^7|a^3=d,b^a=bc,c^a=ce,e^a=eg,c^b=cf,d^b=dg^2123 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 123);