Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24×C3 |
| direct product | C3×D4×C3⋊S4 |
| Fitting | C22×C32×D4 ⋊ D3 |
1314 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 1314 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | C3×S4×C3⋊S4, C3×D3×D4×S4, D3×D4×C3⋊S4, … |
| maximal normal subgroup of | C3×D3×D4×S4, D3×D4×C3⋊S4, D4×#89101 |
| maximal quotient of | D4×A4⋊AGL(1,ℤ/9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47439);