Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #14546466 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C32 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #10624505 |
| commutator | C3×H(3,𝔽3) | C3×C9 |
| Frattini, cosocle | C3×#737 | C32 |
| prime radical | H(5,𝔽3) | C9 |
| LCS | G ▷ C3×H(3,𝔽3) ▷ C33 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C32 ◁ C32×C9 ◁ C3×#737 ◁ G | |
| LECS | G ▷ C3×#737 ▷ C33 ▷ C32 ▷ C3 ▷ C1 | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 243 |
| transitive degree | μt(G) = 243 |
Polycyclic presentation:
3^7|a^3=d,d^3=g^2,b^a=bc,c^a=ce,e^a=ef,f^a=fg,d^b=df^2g,e^b=eg,f^b=fg,d^c=dg^2,e^c=eg^275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 328);