Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #10589519⋊C3 |
302 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 38 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:
2^2,3,2^2,3^2,2^2|b^2=e,d^2=e,f^3=g,b^a=bd,d^a=de,f^a=f^2g^2,g^a=g^2,h^a=hi,d^b=de,f^c=fg,h^f=i,i^f=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12956);