Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3×C9 ⋊ #355 |
268 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^6,3^3|a^2=d,b^2=cd,c^2=f,d^2=f,h^3=i,b^a=bc,c^a=ce,e^a=ef,h^a=h^2i^2,i^a=i^2,c^b=cef,d^b=def,e^b=ef,g^b=g^2,d^c=df108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 1271);