Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of (#126⋊C3)⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #5711⋊C9 |
| unambiguous semidirect product | (#126⋊C9)⋊D3 |
| derived | C3×C42 ⋊ C22×C9 |
258 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^2,3^2,2^2,3,2^2|c^3=d,e^2=h,f^2=hi,e^a=eh,f^a=fhi,g^a=g^2,e^b=ehi,f^b=fi,e^c=ef,f^c=ehi,h^c=i,i^c=hi81 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13288);