Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C3⋊C8×#272 |
| derived, Sylow | C3×H(3,𝔽3) ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3×C4 | D3×C3⋊D3 |
| commutator, hypocenter | C3×H(3,𝔽3) | C2×C8 |
| Fitting, cosocle | C3×C4×H(3,𝔽3) | C22 |
| socle | C2×C32 | C4×C3⋊D3 |
| prime radical | C2×C3×#272 | C4 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C3×C4 | |
330 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1404);