Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D9×C22⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×D9×C22⋊A4 |
| Fitting | C25×C9 ⋊ C2×C3 |
| derived | C24×C9 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9×C22⋊A4 |
| commutator, cosocle, hypocenter | C24×C9 | C22×C3 |
| Frattini | C3 | C2×D3×C22⋊A4 |
| Fitting | C25×C9 | C2×C3 |
| socle | C25×C3 | C3×D3 |
| LCS | G ▷ C24×C9 | |
| UCS | C1 ◁ C2 | |
3028 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 3028 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C22⋊A4×AGL(1,ℤ/9) |
| maximal normal subgroup of | C2×C22⋊A4×AGL(1,ℤ/9) |
| maximal quotient of | C2×D9×#997 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46159);