Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #10588535⋊C3 |
| Sylow | 3−1+2 ⋊ #498 |
312 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 108 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:
2^3,3,2^3,3^2|a^2=f,b^2=f,c^2=fg,e^2=g,h^3=i,b^a=bf,c^a=cf,h^a=h^2i^2,i^a=i^2,c^b=ce,e^b=eg,e^c=eg,h^d=hi^2122 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/9)×#498, #72252828, #72252833 |
| maximal normal subgroup of | AGL(1,ℤ/9)×#498, #72252828, #72252833 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 8217);