Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C9 · Schur multiplier (H2) = C22 |
| direct product | C8×C9×S4 |
| Fitting | C22×C8×C9 ⋊ D3 |
| derived | A4 ⋊ C2×C8×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8×C9 | S4 |
| commutator, hypocenter | A4 | C2×C8×C9 |
| Frattini | C3×C4 | C2×C3×S4 |
| Fitting | C22×C8×C9 | D3 |
| socle | C23×C3 | C3×C4×D3 |
| prime radical | C2×C3×S4 | C3×C4 |
| cosocle | C3×C4×A4 | C22×C3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C8×C9 | |
253 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12826);