Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C26 |
| direct product | C22×D32×C9 |
| unambiguous semidirect product | (C3×D3)⋊(C23×C9) |
| Fitting | C22×C32×C9 ⋊ C22 |
| derived | C32 ⋊ C24×C9 |
| Sylow | C32×C9 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C9 | D32 |
| commutator, hypocenter | C32 | C24×C9 |
| Frattini | C3 | C22×C3×D32 |
| Fitting | C22×C32×C9 | C22 |
| socle | C22×C33 | C22×C3 |
| prime radical | C22×C3×D32 | C3 |
| cosocle | C33 | C24×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22×C9 | |
1000 conjugacy classes of subgroups; only the 312 normal subgroups are shown.
This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 312 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
324 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×D32×D9, C2×C9×#3811 |
| maximal normal subgroup of | C22×D32×D9, C2×C9×#3811 |
| maximal quotient of | C2×C9×#3811 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3031);