Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| direct product | D3×#6189 |
| Sylow | C32×C9 ⋊ C2×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D32×D9 |
| commutator, Fitting, cosocle | C2×C32×C9 | C23 |
| Frattini | C2×C3 | D33 |
| socle | C2×C33 | C22×D3 |
| prime radical | C2×C32×D3 | C2×D3 |
| hypocenter | C32×C9 | C2×Q8 |
| LCS | G ▷ C2×C32×C9 ▷ C32×C9 | |
| UCS | C1 ◁ C2 | |
314 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2000);