Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | ((C22⋊C27)⋊C4)⋊C4 |
275 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 275 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^4,3^3,2^2|a^2=d,b^2=c,e^3=f,f^3=g,b^a=bd,e^a=e^2f^2g^2,f^a=f^2g^2,g^a=g^2,h^a=hi,h^e=i,i^e=hi156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2212);