Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C81 · Schur multiplier (H2) = C2 |
| direct product | C4×C22⋊C81 |
| unambiguous semidirect product | (C22×C4)⋊C81 |
| derived | C22 ⋊ C4×C81 |
| Sylow | C22×C4 ⋊ C81 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C27 | A4 |
| commutator, hypocenter | C22 | C4×C81 |
| Frattini | C2×C27 | C2×A4 |
| Fitting | C22×C4×C27 | C3 |
| socle, prime radical | C23×C3 | C2×C27 |
| cosocle | C23×C27 | C2×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C4×C27 | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 324 |
432 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 60);