Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #14546160 |
| automorphism | ? |
| homology | Abelianization (H1) = C92 · Schur multiplier (H2) = C32 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | #6855 |
| commutator | C3×C9 | C92 |
| Frattini, cosocle | C33×C9 | C32 |
| prime radical | C34 | 3−1+2 |
| LCS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C33×C9 ▷ C32 ▷ C1 | |
272 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 74 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=d,b^3=e,c^3=g^2,e^3=g,b^a=bc,c^a=cf,e^a=eg^2,d^b=dg171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 27);