Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C27 |
| direct product | C23×D3×C22⋊C9 |
| unambiguous semidirect product | C3⋊(C24×C22⋊C9) |
| unambiguous semidirect product | (C2×C3)⋊(C23×C22⋊C9) |
| derived | C22×C3 ⋊ C24×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | D3×A4 |
| commutator, hypocenter | C22×C3 | C24×C9 |
| Frattini | C3 | C23×D3×A4 |
| Fitting, socle | C25×C32 | C2×C3 |
| prime radical | C25×C3×D3 | C3 |
| cosocle | C22×C32 | C24×C3 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C23×C3 | |
4570 conjugacy classes of subgroups; only the 415 normal subgroups are shown.
This group has 4570 conjugacy classes of subgroups — too many to draw. Only its 415 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46157);