Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
| direct product | D4×C3⋊D27 |
| Fitting | C3×D4×C27 ⋊ C2 |
| Sylow | C3×C27 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C3⋊D27 |
| commutator, cosocle | C2×C3×C27 | C23 |
| Frattini | C2×C9 | C22×C3⋊D3 |
| Fitting | C3×D4×C27 | C2 |
| socle | C2×C32 | C22×D9 |
| hypercenter | D4 | C3⋊D27 |
| hypocenter | C3×C27 | C2×D4 |
| LCS | G ▷ C2×C3×C27 ▷ C3×C27 | |
| UCS | C1 ◁ C2 ◁ D4 | |
378 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 324 |
210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1163);