Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D32×D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | D3×(C3⋊D9)⋊C4 |
| unambiguous semidirect product | (D3×C3⋊D9)⋊C4 |
| derived, Sylow | C32×C9 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D32×D9 |
| commutator, hypocenter | C32×C9 | C22×C4 |
| Frattini | C2×C3 | D33 |
| Fitting, cosocle | C2×C32×C9 | C23 |
| socle | C2×C33 | C22×D3 |
| prime radical | C2×D3×C3⋊D9 | C2 |
| LCS | G ▷ C32×C9 | |
| UCS | C1 ◁ C2 | |
464 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#10619528, D3×(C3⋊AGL(1,ℤ/9))⋊C4, D3×#88668 |
| maximal normal subgroup of | D3×(C3⋊AGL(1,ℤ/9))⋊C4, D3×#88668 |
| maximal quotient of | D3×#10619528 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2004);