Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29×C3 |
| direct product | H(5,𝔽2)×(C3⋊D3)⋊C3 |
| Fitting | H(3,𝔽3)×H(5,𝔽2) ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ C2×H(5,𝔽2) |
3309 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
170 conjugacy classes — view the full character table.
| maximal subgroup of | H(5,𝔽2)×PSU(3,𝔽2)⋊C3, H(5,𝔽2)×((C3⋊D3)⋊C3)⋊C2, (C3⋊D3)⋊C3×B(4,𝔽2), … |
| maximal normal subgroup of | H(5,𝔽2)×((C3⋊D3)⋊C3)⋊C2, (C3⋊D3)⋊C3×B(4,𝔽2), #72252896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31014);