Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #6171⋊C8 |
| Fitting | C27×C22⋊C8 ⋊ C2 |
| Sylow | C27 ⋊ C8×D4 |
268 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 864 |
Polycyclic presentation:
2^6,3^3|c^2=d,d^2=f,g^3=h,h^3=i,b^a=be,g^a=g^2h^2i^2,h^a=h^2i^2,i^a=i^2,c^b=ce300 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 296);