Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C27 · Schur multiplier (H2) = C22 |
| direct product | C2×S4×C27 |
| unambiguous semidirect product | A4⋊(C22×C27) |
| unambiguous semidirect product | (C2×A4)⋊(C2×C27) |
| Fitting | C23×C27 ⋊ D3 |
| derived | A4 ⋊ C22×C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C27 | S4 |
| commutator, hypocenter | A4 | C22×C27 |
| Frattini | C9 | C2×C3×S4 |
| Fitting | C23×C27 | D3 |
| socle | C23×C3 | D3×C9 |
| prime radical | C2×C3×S4 | C9 |
| cosocle | C9×A4 | C22×C3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2×C27 | |
144 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 162 |
270 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1815);