Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D3×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C22 |
| direct product | D3×C9×S4 |
| unambiguous semidirect product | (C3×S4)⋊(C2×C9) |
| unambiguous semidirect product | (D3×A4)⋊(C2×C9) |
| Fitting | C22×C3×C9 ⋊ C2×D3 |
| derived | C3×A4 ⋊ C22×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | D3×S4 |
| commutator, hypocenter | C3×A4 | C22×C9 |
| Frattini | C3 | C3×D3×S4 |
| Fitting | C22×C3×C9 | C2×D3 |
| socle | C22×C32 | C2×C3×D3 |
| prime radical | C3×D3×S4 | C3 |
| cosocle | C32×A4 | C22×C3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C9 | |
256 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 72 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | S4×D9⋊C9 |
| maximal quotient of | S4×D9⋊C9 |
| automorphism group of | A4×C27, C22×C22⋊C27 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2918);