Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C8 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | #504⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×#504⋊C3)⋊C3 |
| Fitting | C32×#504 ⋊ C3 |
| Sylow | #504 ⋊ H(3,𝔽3) |
165 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 165 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2,3^2,2,3,2^4|a^2=d,d^2=f,f^2=i,g^2=i,h^2=i,c^b=ce,g^b=gh,h^b=g,h^g=hi248 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7330);