Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #14665⋊C3 |
| Fitting, Sylow | #14665 ⋊ C3 |
340 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 38 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=f,c^2=i,e^2=fg,b^a=bei,d^a=df,e^a=bfgi,f^a=g,g^a=fg,c^b=cf,d^b=dg,d^c=dh,e^c=efg,e^d=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | #72258627, #72258641, #72258809, … |
| maximal normal subgroup of | #72258627, #72258641, #72258809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083645);