Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C9 · Schur multiplier (H2) = C43 |
| direct product | C42×C9×Dic3 |
| unambiguous semidirect product | C3⋊(C43×C9) |
| derived | C3 ⋊ C43×C9 |
| Sylow | C3×C9 ⋊ C43 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C42×C9 | D3 |
| commutator, hypocenter | C3 | C43×C9 |
| Frattini | C23×C3 | C22×C3×D3 |
| Fitting | C2×C3×C42×C9 | C2 |
| socle, prime radical, cosocle | C23×C32 | C23×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C42×C9 | |
882 conjugacy classes of subgroups; only the 549 normal subgroups are shown.
This group has 882 conjugacy classes of subgroups — too many to draw. Only its 549 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
864 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 8453);