Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C9 · Schur multiplier (H2) = C2×C4 |
| direct product | C4×Dic3×C22⋊C9 |
| unambiguous semidirect product | C3⋊(C42×C22⋊C9) |
| unambiguous semidirect product | (C22×C4×Dic3)⋊C9 |
| derived | C22×C3 ⋊ C42×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | D3×A4 |
| commutator, hypocenter | C22×C3 | C42×C9 |
| Frattini | C22×C3 | C2×D3×A4 |
| Fitting | C23×C32×C4 | C2×C3 |
| socle, prime radical, cosocle | C24×C32 | C22×C3 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C2×C3×C4 | |
538 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 538 conjugacy classes of subgroups — too many to draw. Only its 115 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, 13321);