Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C9×C16 · Schur multiplier (H2) = C2 |
| direct product | C9×A4⋊C16 |
| unambiguous semidirect product | A4⋊(C9×C16) |
| derived | A4 ⋊ C9×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8×C9 | S4 |
| commutator, hypocenter | A4 | C9×C16 |
| Frattini | C3×C8 | C3×S4 |
| Fitting | C22×C8×C9 | D3 |
| socle | C23×C3 | C3×C3⋊C8 |
| prime radical | C2×C3×A4 | C3×C8 |
| cosocle | C3×C8×A4 | C2×C3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C8×C9 | |
115 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 115 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 432 |
360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2808);