Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #32708⋊C3 |
| Fitting, Sylow | #32708 ⋊ C3 |
232 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 24 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|c^2=i,h^2=i,b^a=befgi,c^a=cfg,d^a=dg,e^a=bf,f^a=g,g^a=fg,c^b=cf,d^b=dg,d^c=dh,e^c=efg,h^c=hi,e^d=ef,h^d=hi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083948);