Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | #10589382⋊C3 |
| Fitting | H(3,𝔽3)×M32 ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #361 |
243 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 60 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:
2^2,3,2^4,3^2|b^2=d,d^2=fg,e^2=fg,f^2=g,b^a=be,d^a=df,e^a=ef,f^a=fg,h^a=h^2,i^a=i^2,e^b=eg,h^c=hi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 1576);