Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | #557⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×#557⋊C3)⋊C3 |
| Fitting | C32×#557 ⋊ C3 |
| Sylow | #557 ⋊ H(3,𝔽3) |
371 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 371 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^2,3^2,2,3,2^3|a^2=e,b^2=ei,g^2=i,h^2=i,b^a=be,d^c=df,g^c=gh,h^c=g,h^g=hi155 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18264);