Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×#10590082 |
| Sylow | C33 ⋊ C4⋊C16 |
130 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C16×PSU(3,𝔽2), C3×GO+(4,𝔽2)⋊C16, C3×AGL(1,𝔽9)⋊C16 |
| maximal normal subgroup of | C3×C16×PSU(3,𝔽2), C3×GO+(4,𝔽2)⋊C16, C3×AGL(1,𝔽9)⋊C16 |
| maximal quotient of | C3×#11773018 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 21189);