Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C3×C4 |
| direct product | #127×(C3⋊D3)⋊C3 |
| Fitting | H(3,𝔽3)×#127 ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ C2×#127 |
1851 conjugacy classes of subgroups; only the 288 normal subgroups are shown.
This group has 1851 conjugacy classes of subgroups — too many to draw. Only its 288 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
140 conjugacy classes — view the full character table.
| maximal subgroup of | #127×PSU(3,𝔽2)⋊C3, #127×((C3⋊D3)⋊C3)⋊C2, (C3⋊D3)⋊C3×#453 |
| maximal normal subgroup of | #127×((C3⋊D3)⋊C3)⋊C2, (C3⋊D3)⋊C3×#453 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 19336);