Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C3⋊PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C3×C3⋊PSU(3,𝔽2) |
| unambiguous semidirect product | (C2×C3)⋊(C3×PSU(3,𝔽2)) |
| Fitting, socle | C2×C34 ⋊ Q8 |
| Sylow | C34 ⋊ C2×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊PSU(3,𝔽2) |
| commutator, cosocle | C3×C3⋊D3 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C34 | Q8 |
| prime radical | C2×C32×C3⋊D3 | C22 |
| hypocenter | C33 | C2×C3×Q8 |
| LCS | G ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2×C3 | |
242 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×D3×PSU(3,𝔽2), C2×D3×C3⋊PSU(3,𝔽2), C2×#13049817, … |
| maximal normal subgroup of | C2×C3×D3×PSU(3,𝔽2), C2×D3×C3⋊PSU(3,𝔽2), C2×#13049817, … |
| maximal quotient of | C3×#11724369 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3578);