Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C22×C32 |
| unambiguous semidirect product | C32⋊#5465 |
| Fitting | C32×H(3,𝔽4) ⋊ C3 |
| Sylow | H(3,𝔽4) ⋊ H(3,𝔽3) |
458 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 37 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:
3^3,2^6|b^a=bc,d^a=de,e^a=d,f^a=fg,g^a=f,f^d=fi,g^d=ghi,f^e=fh,g^e=gi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18330);