Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C3×C4 |
| unambiguous semidirect product | #10592773⋊C3 |
| Fitting | H(3,𝔽3)×#127 ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #496 |
612 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 100 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|a^2=f,f^2=g,b^a=bfg,c^a=cf,h^a=h^2,i^a=i^2,c^b=ce,f^b=fg,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, 9044);