Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C27 · Schur multiplier (H2) = C22×C3 |
| direct product | C22×C3×C22⋊C27 |
| derived | C22 ⋊ C22×C3×C27 |
| Sylow | C24 ⋊ C3×C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3×C9 | A4 |
| commutator, hypocenter | C22 | C22×C3×C27 |
| Frattini | C9 | C22×C3×A4 |
| Fitting | C24×C3×C9 | C3 |
| socle, prime radical | C24×C32 | C9 |
| cosocle | C22×C9 | C22×C32 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C22×C3×C9 | |
330 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 324 |
432 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1865);