Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of AΓL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×AΓL(1,𝔽9) |
| unambiguous semidirect product | AGL(1,𝔽9)⋊(C2×C5) |
| unambiguous semidirect product | PSU(3,𝔽2)⋊(C2×C5) |
| Fitting, socle, Hall | C32×C5 ⋊ SD16 |
| Sylow | C32 ⋊ C5×SD16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | AΓL(1,𝔽9) |
| commutator, cosocle | #157 | C22×C5 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5 | SD16 |
| prime radical | C5×GO+(4,𝔽2) | C2 |
| hypocenter | C32 | C5×SD16 |
| LCS | G ▷ #157 ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C5 | |
50 conjugacy classes of subgroups.
| transitive degree | μt(G) = 45 |
45 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 779);