Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C22⋊C4×((C3⋊D3)⋊C3)⋊C2 |
| Fitting | C22⋊C4×H(3,𝔽3) ⋊ C22 |
| Sylow | H(3,𝔽3) ⋊ C22×C22⋊C4 |
2491 conjugacy classes of subgroups; only the 267 normal subgroups are shown.
This group has 2491 conjugacy classes of subgroups — too many to draw. Only its 267 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 | C22⋊C4×AGL(2,𝔽3), #120×((C3⋊D3)⋊C3)⋊C2, #99×((C3⋊D3)⋊C3)⋊C2 |
| maximal normal subgroup of | #120×((C3⋊D3)⋊C3)⋊C2, #99×((C3⋊D3)⋊C3)⋊C2 |
| maximal quotient of | #99×((C3⋊D3)⋊C3)⋊C2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 14768);