Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C81 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C9⋊C81 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C3×C27 | C32 |
| commutator | C3 | C3×C81 |
| socle, prime radical | C32 | C3×C27 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C27 ◁ G | |
| LECS | G ▷ C3×C27 ▷ C9 ▷ C3 ▷ C1 | |
35 conjugacy classes of subgroups.
| transitive degree | μt(G) = 729 |
Polycyclic presentation:
3^6|a^3=d,b^3=c,d^3=e,e^3=f,b^a=bc297 conjugacy classes — view the full character table.
| Fitting subgroup of | D9⋊C81 |
| maximal subgroup of | C2×C9⋊C81, D9⋊C81, #14546334, … |
| maximal normal subgroup of | C2×C9⋊C81, D9⋊C81, #14546334, … |
| maximal quotient of | C2×C9⋊C81, #14546334, #14546336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 91);