Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #10588817⋊C3 |
| Fitting | C2×M16×3−1+2 ⋊ C2 |
| Sylow | 3−1+2 ⋊ #469 |
424 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^3,3,2^3,3^2|b^2=fg,f^2=g,h^3=i,b^a=bf,c^a=ce,f^a=fg,h^a=h^2i^2,i^a=i^2,c^b=cg,h^d=hi^2128 conjugacy classes — view the full character table.
| maximal subgroup of | #72252812, AGL(1,ℤ/9)×#469, #72252833 |
| maximal normal subgroup of | #72252812, AGL(1,ℤ/9)×#469, #72252833 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 9304);