Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #14551662 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C32 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | #6911 |
| commutator, Frattini, cosocle | C32×C9 | C33 |
| prime radical | C34 | H(3,𝔽3) |
| LCS | G ▷ C32×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C33 ◁ C32×3−1+2 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C32 ▷ C3 ▷ C1 | |
204 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 47 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,b^3=g^2,c^3=df^2,e^3=g^2,b^a=bd,c^a=ce,d^a=dg,e^a=ef,f^a=fg,e^b=eg,d^c=dg,e^c=ef,f^c=fg91 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 5840);