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×A4×C27 |
| Fitting | C24×C27 ⋊ C3 |
| derived | C22 ⋊ C22×C3×C27 |
| Sylow | C24 ⋊ C3×C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C27 | A4 |
| commutator, hypocenter | C22 | C22×C3×C27 |
| Frattini | C9 | C22×C3×A4 |
| Fitting | C24×C27 | C3 |
| socle | C24×C3 | C3×C9 |
| prime radical | C22×C3×A4 | C9 |
| cosocle | C22×C9 | C22×C32 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C22×C27 | |
216 conjugacy classes of subgroups.
| 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, 1863);