Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×D3×#272 |
| Fitting | C22×C3×H(3,𝔽3) ⋊ C22 |
| derived, Sylow | C3×H(3,𝔽3) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D3×C3⋊D3 |
| commutator, cosocle, hypocenter | C3×H(3,𝔽3) | C24 |
| Frattini | C3 | C22×D3×C3⋊D3 |
| Fitting | C22×C3×H(3,𝔽3) | C22 |
| socle | C22×C32 | C2×C3⋊D3 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C22×C3 | |
1746 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 1746 conjugacy classes of subgroups — too many to draw. Only its 209 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, 3038);