Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #21666⋊C3 |
| Fitting, Sylow | #21666 ⋊ C3 |
172 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 172 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=h,c^2=fg,d^2=f,e^2=h,h^2=i,b^a=bfg,c^a=dfi,d^a=cdf,e^a=ef,f^a=g,g^a=fg,c^b=cf,d^b=dg,e^b=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083863);