Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C22×Dic3×C22⋊C9 |
| derived | C22×C3 ⋊ C22×C4×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | D3×A4 |
| commutator, hypocenter | C22×C3 | C22×C4×C9 |
| Frattini | C2×C3 | C22×D3×A4 |
| Fitting, socle, prime radical | C25×C32 | C2×C3 |
| cosocle | C23×C32 | C23×C3 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C23×C3 | |
1338 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 1338 conjugacy classes of subgroups — too many to draw. Only its 215 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, 30704);