Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #10588126⋊C3 |
| Fitting | 3−1+2×C32 ⋊ C2 |
| Sylow | 3−1+2 ⋊ SD64 |
102 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 102 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.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^2,3,2^4,3^2|b^2=deg,d^2=ef,e^2=fg,f^2=g,h^3=i,b^a=bd,d^a=de,e^a=ef,f^a=fg,h^a=h^2i^2,i^a=i^2,h^c=hi^2185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 1312);