Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C3 |
| direct product | D4×C9⋊S4 |
| Fitting | C22×D4×C9 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C9⋊S4 |
| commutator, cosocle | C2×C9×A4 | C23 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting | C22×D4×C9 | D3 |
| socle | C23×C3 | C22×C3⋊D3 |
| hypercenter | D4 | C9⋊S4 |
| hypocenter | C9×A4 | C2×D4 |
| LCS | G ▷ C2×C9×A4 ▷ C9×A4 | |
| UCS | C1 ◁ C2 ◁ D4 | |
918 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 918 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | D4×A4⋊AGL(1,ℤ/9) |
| maximal normal subgroup of | D4×A4⋊AGL(1,ℤ/9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30516);