Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of AΓL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×PSU(3,𝔽2)⋊C8 |
| Sylow | C33 ⋊ Q8⋊C8 |
170 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | C3×(PSU(3,𝔽2)⋊C3)⋊C8, C3×C8×AΓL(1,𝔽9) |
| maximal normal subgroup of | C3×C8×AΓL(1,𝔽9) |
| maximal quotient of | C3×PSU(3,𝔽2)⋊C16 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31134);