Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | #99×AGL(1,ℤ/9) |
| unambiguous semidirect product | D9⋊(C3×#99) |
| unambiguous semidirect product | (D9×#99)⋊C3 |
| Fitting | 3−1+2×#99 ⋊ C2 |
| Sylow | 3−1+2 ⋊ C2×#99 |
656 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
110 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/9)×B(4,𝔽2), #72220150, #72220157, … |
| maximal normal subgroup of | AGL(1,ℤ/9)×B(4,𝔽2), C2×#99×AGL(1,ℤ/9) |
| maximal quotient of | #72220150, #72220157, C2×#99×AGL(1,ℤ/9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7722);