Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of C3⋊PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C3 |
| direct product | C3×C3⋊PSU(3,𝔽2) |
| unambiguous semidirect product | C3⋊(C3×PSU(3,𝔽2)) |
| Fitting, socle, Sylow | C34 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊PSU(3,𝔽2) |
| commutator, cosocle | C3×C3⋊D3 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C34 | Q8 |
| prime radical | C32×C3⋊D3 | C22 |
| hypocenter | C33 | C3×Q8 |
| LCS | G ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C3 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 24 |
36 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#89091, C3×AGL(1,𝔽9)⋊D3, #13049797, … |
| maximal normal subgroup of | C3×#89091, C3×AGL(1,𝔽9)⋊D3, #13049797, … |
| maximal quotient of | C3×#88945, C3×#88946, C2×C3×C3⋊PSU(3,𝔽2), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 727);