Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | #14686⋊C3 |
| Fitting, Sylow | #14686 ⋊ C3 |
538 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 538 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=i,c^2=f,d^2=g,b^a=bcd,c^a=cd,d^a=cg,e^a=eg,f^a=fg,g^a=f,c^b=cf,d^b=dg,e^b=eh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083652);