Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C23 |
| direct product | C22×C9×Dic9 |
| derived | C9 ⋊ C22×C4×C9 |
| Sylow | C92 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C9 | D9 |
| commutator, hypocenter | C9 | C22×C4×C9 |
| Frattini | C2×C32 | C22×C3×D3 |
| Fitting | C23×C92 | C2 |
| socle, prime radical | C23×C32 | C3×D3 |
| cosocle | C2×C3×C9 | C23×C3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C23×C9 | |
355 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 177 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
432 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C9×#3699, C2×#13047153, C22×#10619528, … |
| maximal normal subgroup of | C2×C9×#3699, C2×#13047153, C22×#10619528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1105);