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 | D9×#624 |
| 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×D9×C3⋊D3 | C2 |
| LCS | G ▷ C32×C9 | |
| UCS | C1 ◁ C2 | |
456 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 80 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 | AGL(1,ℤ/9)×#624, D9×#3811, C2×D9×#624 |
| maximal normal subgroup of | AGL(1,ℤ/9)×#624, D9×#3811, C2×D9×#624 |
| maximal quotient of | C2×D9×#624 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2003);