Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #10624480 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C33 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C33 | #732 |
| commutator | C32×C9 | C3×C9 |
| Frattini, cosocle | C33×C9 | C32 |
| prime radical | C34 | H(3,𝔽3) |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ C34 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C33×C9 ▷ C33 ▷ C3 ▷ C1 | |
383 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 383 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|a^3=d,b^3=g,c^3=g^2,b^a=bc,c^a=ce,e^a=eg,f^a=fg,c^b=cf,e^b=eg,f^b=fg171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 133);