Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C23×C32×C4 |
| unambiguous semidirect product | C32⋊(C22⋊(C42⋊C3)) |
| Fitting | C22×C32×C42 ⋊ C3 |
| Sylow | C22×C42 ⋊ H(3,𝔽3) |
606 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
3^3,2^6|f^2=h,g^2=hi,b^a=bc,d^a=de,e^a=d,f^a=fg,g^a=fhi,h^a=i,i^a=hi200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18318);