Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C3 |
| direct product | C9×PSU(3,𝔽2) |
| Fitting, Sylow | C32×C9 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | PSU(3,𝔽2) |
| commutator | C3⋊D3 | C22×C9 |
| Frattini | C3 | C3×PSU(3,𝔽2) |
| Fitting | C32×C9 | Q8 |
| socle | C33 | C3×Q8 |
| prime radical, cosocle | C3×C3⋊D3 | C22×C3 |
| hypocenter | C32 | Q8×C9 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C9 | |
46 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | C9×AΓL(1,𝔽9), #13049185, D9×PSU(3,𝔽2), … |
| maximal normal subgroup of | C9×AΓL(1,𝔽9), #13049185, D9×PSU(3,𝔽2), … |
| maximal quotient of | C9×#3777, C2×C9×PSU(3,𝔽2), C9×SU(3,𝔽2), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 549);