Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D3×C3⋊PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
| direct product | D3×C3⋊PSU(3,𝔽2) |
| unambiguous semidirect product | (C3×D3)⋊PSU(3,𝔽2) |
| Fitting, socle, Sylow | C34 ⋊ C2×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | C32×C3⋊D3 | C23 |
| Fitting, socle, hypocenter | C34 | C2×Q8 |
| prime radical | C3×D3×C3⋊D3 | C22 |
| LCS | G ▷ C32×C3⋊D3 ▷ C34 | |
| UCS | C1 | |
209 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
36 conjugacy classes — view the full character table.
| maximal subgroup of | D32×PSU(3,𝔽2), D3×#89091, D3×AGL(1,𝔽9)⋊D3, … |
| maximal normal subgroup of | D32×PSU(3,𝔽2), D3×#89091, D3×AGL(1,𝔽9)⋊D3, … |
| maximal quotient of | C2×D3×C3⋊PSU(3,𝔽2), #72259504, #72259532, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3533);