Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of AΓL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×GO+(4,𝔽2)⋊C4 |
| unambiguous semidirect product | GO+(4,𝔽2)⋊(C2×C3×C4) |
| Sylow | C33 ⋊ C2×#102 |
570 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 570 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×GO+(4,𝔽2)⋊AGL(1,𝔽5), C2×C3×C4×AΓL(1,𝔽9) |
| maximal normal subgroup of | C2×C3×C4×AΓL(1,𝔽9) |
| maximal quotient of | C2×C3×GO+(4,𝔽2)⋊AGL(1,𝔽5) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46451);