Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | ((C2×D9)⋊C4)⋊A4 |
| unambiguous semidirect product | (C2×D9⋊A4)⋊C4 |
| Fitting | C23×C4×C9 ⋊ C2×C3 |
523 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 523 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^2,3,2^4,3^2|b^2=d,h^3=i,b^a=be,h^a=h^2i^2,i^a=i^2,f^c=fg,g^c=f,h^c=hi^2104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13220);