Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #10588131⋊C3 |
| Sylow | 3−1+2 ⋊ #322 |
220 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^2,3,2^4,3^2|a^2=e,b^2=d,d^2=f,e^2=g,h^3=i,b^a=bg,h^a=h^2i^2,i^a=i^2,h^c=hi^2248 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/9)×#322, #72245408, #72252773, … |
| maximal normal subgroup of | AGL(1,ℤ/9)×#322, #72245408, #72252773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 1357);