Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #17520⋊C3 |
| Fitting, Sylow | #17520 ⋊ C3 |
274 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
3,2^8|b^2=g,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,d^a=eh,e^a=degh,c^b=cf,f^b=fh,f^c=fh,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083750);