Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×C8×D9 |
| Fitting | C32×C8×C9 ⋊ C2 |
| derived | C9 ⋊ C2×C32×C8 |
| Sylow | C32×C9 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C8 | D9 |
| commutator, hypocenter | C9 | C2×C32×C8 |
| Frattini | C3×C4 | C2×C32×D3 |
| Fitting | C32×C8×C9 | C2 |
| socle | C2×C33 | C4×D3 |
| prime radical | C2×C32×D9 | C4 |
| cosocle | C4×C9 | C22×C32 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C32×C8 | |
262 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
432 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1508);