Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C32 |
| direct product | C32×PSU(3,𝔽2) |
| Fitting, socle, Sylow | C34 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | PSU(3,𝔽2) |
| commutator, cosocle | C3⋊D3 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C34 | Q8 |
| prime radical | C32×C3⋊D3 | C22 |
| hypocenter | C32 | C32×Q8 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C32 | |
106 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 726);