Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3×D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
| direct product | C2×D9×C3⋊C16 |
| derived, Sylow | C3×C9 ⋊ C22×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | D3×D9 |
| commutator, hypocenter | C3×C9 | C22×C16 |
| Frattini | C3×C8 | C2×D32 |
| Fitting | C2×C3×C8×C9 | C22 |
| socle | C22×C32 | D3×C8 |
| prime radical | C22×C3×D9 | C8 |
| cosocle | C3×C8×C9 | C23 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C2×C8 | |
322 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 4132);