Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C9⋊(Q8⋊SL(2,𝔽3)) |
| Fitting | Q82×C9 ⋊ C3 |
| Sylow | Q82 ⋊ 3−1+2 |
218 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
3^3,2^6|b^3=c,d^2=i,e^2=i,f^2=h,g^2=h,b^a=bc,d^a=de,e^a=d,f^a=fg,g^a=f,e^d=ei,g^f=gh107 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18309);