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) = C1 |
| unambiguous semidirect product | C243⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C81 | C32 |
| commutator, socle | C3 | C3×C81 |
| prime radical | C32 | C81 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C81 ◁ G | |
| LECS | G ▷ C81 ▷ C27 ▷ C9 ▷ C3 ▷ C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 243 |
| transitive degree | μt(G) = 243 |
Polycyclic presentation:
3^6|a^3=c,c^3=d,d^3=e,e^3=f,b^a=bf297 conjugacy classes — view the full character table.
| Fitting subgroup of | D243⋊C3 |
| maximal subgroup of | C2×C243⋊C3, D243⋊C3, #14546455, … |
| maximal normal subgroup of | C2×C243⋊C3, D243⋊C3, #14546455, … |
| maximal quotient of | C2×C243⋊C3, #14546454, C32⋊C243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 94);