Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9 · Schur multiplier (H2) = C35×C9 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C32×C9 | C33 |
| commutator | C32 | C33×C9 |
| Frattini, socle, cosocle | C33 | C34 |
| prime radical | C35 | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C1 | |
2555 conjugacy classes of subgroups; only the 656 normal subgroups are shown.
This group has 2555 conjugacy classes of subgroups — too many to draw. Only its 656 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 243 |
Polycyclic presentation:
3^7|a^3=g,d^3=f,b^a=be,c^a=cf459 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 5910);