Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | 3−1+2⋊#338 |
| unambiguous semidirect product | (C9⋊#338)⋊C3 |
| Sylow | 3−1+2 ⋊ #338 |
292 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^2,3,2^4,3^2|a^2=f,b^2=d,d^2=g,e^2=g,f^2=g,h^3=i,b^a=be,d^a=dg,h^a=h^2i^2,i^a=i^2,f^b=fg,h^c=hi^2146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 1749);