Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #6204⋊C8 |
| unambiguous semidirect product | ((Q8⋊C9)⋊C8)⋊C3 |
| derived | Q8⋊C9 ⋊ C3×C8 |
157 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2,3,2^2,3^2,2^3|a^2=c,c^2=d,e^3=f,g^2=i,h^2=i,e^a=e^2f^2,f^a=f^2,g^a=gh,h^a=hi,e^b=ef,g^e=h,h^e=ghi,h^g=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2820);