Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×#543⋊C9 |
| unambiguous semidirect product | (C3×#543)⋊C9 |
| derived, Sylow | #543 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C32 | C22⋊A4 |
| commutator, hypocenter | #543 | C3×C9 |
| Frattini | C22×C3 | C3×C22⋊A4 |
| Fitting | C32×#543 | C3 |
| prime radical | C24×C32 | A4 |
| cosocle | C3×#543 | C32 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C22×C32 | |
394 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18319);