Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #17419⋊C3 |
| Fitting, Sylow | #17419 ⋊ C3 |
332 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
3,2^8|b^2=g,c^2=h,d^2=i,e^2=i,g^2=i,d^a=ehi,e^a=dehi,c^b=cf,f^b=fh,g^c=gh,e^d=ei77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083740);