Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #10624480 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C32 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | #732 |
| commutator, Frattini, cosocle | C32×C9 | C33 |
| socle | C32 | C3×#732 |
| prime radical | C3×#6908 | C3 |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ C32×C9 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
321 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 62 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|b^3=e^2,c^3=g,d^3=g^2,b^a=bd,d^a=de,e^a=eg,d^b=df171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 4703);