Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #14302⋊C3 |
| Fitting, Sylow | #14302 ⋊ C3 |
339 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
3,2^8|b^2=i,c^2=f,d^2=f,e^2=fi,b^a=bcfghi,c^a=beg,d^a=dh,e^a=efg,g^a=gh,h^a=fg,c^b=cf,d^b=dg,d^c=dh,e^d=ef48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083620);