Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #10588598⋊C3 |
| Fitting | 3−1+2×Q32 ⋊ C2 |
| Sylow | 3−1+2 ⋊ #510 |
258 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^3,3,2^3,3^2|b^2=g,c^2=g,e^2=fg,f^2=g,h^3=i,c^a=cg,h^a=h^2i^2,i^a=i^2,c^b=ce,e^b=ef,f^b=fg,e^c=ef,f^c=fg,h^d=hi^292 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 8425);