Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | ((C22⋊C9)⋊C4)⋊Dic3 |
447 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 447 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) = 144 |
Polycyclic presentation:
2^4,3^3,2^2|a^2=d,b^2=c,e^3=g,b^a=bc,f^a=f^2,e^b=e^2g^2,g^b=g^2,h^b=hi,h^e=i,i^e=hi96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12456);