Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22×C32 |
| unambiguous semidirect product | #393⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×#393⋊C3)⋊C3 |
| Fitting | C32×#393 ⋊ C3 |
| Sylow | #393 ⋊ H(3,𝔽3) |
328 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3^3,2^5|e^2=i,f^2=i,e^a=eg,f^a=fgh,c^b=cd,e^b=ef,f^b=e,g^b=h,h^b=gh,f^e=fi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7303);