Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24×C3 |
| direct product | #99×(C3⋊D3)⋊C3 |
| unambiguous semidirect product | (C3⋊D3×#99)⋊C3 |
| Fitting | H(3,𝔽3)×#99 ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ C2×#99 |
787 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 787 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
110 conjugacy classes — view the full character table.
| maximal subgroup of | #99×PSU(3,𝔽2)⋊C3, (C3⋊D3)⋊C3×B(4,𝔽2), #72252984, … |
| maximal normal subgroup of | (C3⋊D3)⋊C3×B(4,𝔽2), #72252984, #99×((C3⋊D3)⋊C3)⋊C2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7718);