Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #10588750⋊C3 |
| Sylow | 3−1+2 ⋊ #486 |
408 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 408 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) = 288 |
Polycyclic presentation:
2^3,3,2^3,3^2|a^2=f,b^2=f,e^2=g,h^3=i,b^a=bf,c^a=ce,e^a=eg,h^a=h^2i^2,i^a=i^2,c^b=cf,e^c=eg,h^d=hi^2122 conjugacy classes — view the full character table.
| maximal subgroup of | #72245382, AGL(1,ℤ/9)×#486, #72252773, … |
| maximal normal subgroup of | #72245382, AGL(1,ℤ/9)×#486, #72252773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 9054);