Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C32×C4 |
| unambiguous semidirect product | #384⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×#384⋊C3)⋊C3 |
| Fitting | C32×#384 ⋊ C3 |
| Sylow | #384 ⋊ H(3,𝔽3) |
267 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 36 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:
2,3^2,2,3,2^4|a^2=d,f^2=h,g^2=hi,f^a=fh,g^a=ghi,c^b=ce,f^b=fg,g^b=fhi,h^b=i,i^b=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7297);