Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | #10592876⋊C3 |
| Fitting | C2×SD16×H(3,𝔽3) ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #471 |
432 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 86 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:
2^3,3,2^3,3^2|c^2=g,e^2=g,f^2=g,b^a=bf,c^a=cf,f^a=fg,h^a=h^2,i^a=i^2,c^b=ce,e^b=eg,f^b=fg,e^c=eg,f^c=fg,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 9441);