Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C22⋊#401)⋊C3 |
| Fitting, Sylow | C22⋊#401 ⋊ C3 |
345 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 31 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=fgh,c^2=fg,d^2=f,b^a=bcdg,c^a=cdfghi,d^a=cfgh,e^a=eg,f^a=fg,g^a=fh,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, 1083663);