Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C27 · Schur multiplier (H2) = C22 |
| direct product | C2×C8×C22⋊C27 |
| unambiguous semidirect product | (C23×C8)⋊C27 |
| derived | C22 ⋊ C2×C8×C27 |
| Sylow | C23×C8 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8×C9 | A4 |
| commutator, hypocenter | C22 | C2×C8×C27 |
| Frattini | C4×C9 | C22×A4 |
| Fitting | C23×C8×C9 | C3 |
| socle, prime radical | C24×C3 | C4×C9 |
| cosocle | C22×C4×C9 | C22×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C2×C8×C9 | |
235 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
576 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2252);