Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×C8×PSU(3,𝔽2) |
| Fitting | C33×C8 ⋊ Q8 |
| Sylow | C33 ⋊ C8×Q8 |
234 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 98 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×C8×PSU(3,𝔽2)⋊C3, C3×C16×PSU(3,𝔽2), C3×C8×AΓL(1,𝔽9), … |
| maximal normal subgroup of | C3×C8×PSU(3,𝔽2)⋊C3, C3×C16×PSU(3,𝔽2), C3×C8×AΓL(1,𝔽9), … |
| maximal quotient of | C3×C16×PSU(3,𝔽2) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 33670);