Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C9×#393⋊C3 |
| Fitting | C9×#393 ⋊ C3 |
| derived | C22×Q8 ⋊ C2×C3×C9 |
| Sylow | #393 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | #120⋊C3 |
| commutator, hypocenter | C22×Q8 | C2×C3×C9 |
| Frattini, socle | C23×C3 | C2×C3×A4 |
| Fitting | C9×#393 | C3 |
| prime radical | C3×#393⋊C3 | C3 |
| cosocle | C22×C3×Q8 | C2×C32 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2×C9 | |
227 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7258);