Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | D4⋊(C3⋊#622) |
| Sylow | C33 ⋊ D4⋊C8 |
460 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|a^2=c,c^2=f,d^2=e,b^a=bd,d^a=de,g^a=g^2,h^a=hi,i^a=hi^2,d^b=de,h^c=h^2,i^c=i^2108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 23293);