Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C9 · Schur multiplier (H2) = C26×C3 |
| direct product | C23×C3×D3×C9 |
| unambiguous semidirect product | C3⋊(C24×C3×C9) |
| unambiguous semidirect product | (C2×C3)⋊(C23×C3×C9) |
| Fitting | C23×C32×C9 ⋊ C2 |
| derived | C3 ⋊ C24×C3×C9 |
| Sylow | C32×C9 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3×C9 | D3 |
| commutator, hypocenter | C3 | C24×C3×C9 |
| Frattini | C3 | C23×C32×D3 |
| Fitting | C23×C32×C9 | C2 |
| socle | C23×C33 | C2×C3 |
| prime radical | C23×C32×D3 | C3 |
| cosocle | C32 | C24×C32 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C23×C3×C9 | |
1580 conjugacy classes of subgroups; only the 830 normal subgroups are shown.
This group has 1580 conjugacy classes of subgroups — too many to draw. Only its 830 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
648 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3063);