Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #10588532⋊C3 |
| Fitting | 3−1+2×#106 ⋊ C2 |
| Sylow | 3−1+2 ⋊ C4×SD16 |
384 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^3,3,2^3,3^2|b^2=f,c^2=e,f^2=g,h^3=i,b^a=bf,f^a=fg,h^a=h^2i^2,i^a=i^2,c^b=cfg,f^c=fg,h^d=hi^2140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 8234);