Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #15214⋊C3 |
| Fitting, Sylow | #15214 ⋊ C3 |
331 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 331 conjugacy classes of subgroups — too many to draw. Only its 29 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=h,d^2=h,e^2=hi,b^a=bg,c^a=cdfh,d^a=cfghi,f^a=fg,g^a=f,c^b=cf,d^b=dg,e^b=ei,d^c=dh48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083690);