Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #10588427⋊C3 |
| Fitting | 3−1+2×#100 ⋊ C2 |
| Sylow | 3−1+2 ⋊ #453 |
527 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 527 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3,2^3,3^2|c^2=fg,f^2=g,h^3=i,c^a=cf,e^a=eg,f^a=fg,h^a=h^2i^2,i^a=i^2,c^b=ce,f^b=fg,e^c=eg,h^d=hi^292 conjugacy classes — view the full character table.
| maximal subgroup of | #72220032, AGL(1,ℤ/9)×#453, C2×#10588427⋊C3 |
| maximal normal subgroup of | AGL(1,ℤ/9)×#453, C2×#10588427⋊C3 |
| maximal quotient of | #72220032, C2×#10588427⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7744);