Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24×C3 |
| unambiguous semidirect product | #10592440⋊C3 |
| Fitting | H(3,𝔽3)×#99 ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ B(4,𝔽2) |
745 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 745 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3,2^3,3^2|c^2=f,c^a=cf,e^a=eg,h^a=h^2,i^a=i^2,c^b=ce,f^b=fg,e^c=eg,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | (C3⋊D3)⋊C3×B(4,𝔽2), #72252984, #72253278 |
| maximal normal subgroup of | (C3⋊D3)⋊C3×B(4,𝔽2), #72252984, #72253278 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7716);