Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C9×#543⋊C3 |
| Fitting | C9×#543 ⋊ C3 |
| derived, Sylow | #543 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C9 | C22⋊A4 |
| commutator, hypocenter | #543 | C3×C9 |
| Frattini, socle | C22×C3 | C3×C22⋊A4 |
| Fitting | C9×#543 | C3 |
| prime radical | C3×#543⋊C3 | C3 |
| cosocle | C3×#543 | C32 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C22×C9 | |
292 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18306);