Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D3×D27 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×D3×D27 |
| unambiguous semidirect product | (C3×D27)⋊C23 |
| unambiguous semidirect product | (D3×C27)⋊C23 |
| unambiguous semidirect product | (C3⋊D27)⋊C23 |
| Fitting | C22×C3×C27 ⋊ C22 |
| derived, Sylow | C3×C27 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×D27 |
| commutator, cosocle, hypocenter | C3×C27 | C24 |
| Frattini | C9 | C22×D32 |
| Fitting | C22×C3×C27 | C22 |
| socle | C22×C32 | C2×D9 |
| LCS | G ▷ C3×C27 | |
| UCS | C1 ◁ C22 | |
584 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1859);