Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C4⋊(#376⋊C3) |
| unambiguous semidirect product | #13810⋊C3 |
| Fitting, Sylow | #13810 ⋊ C3 |
329 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 34 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=h,c^2=i,d^2=f,e^2=fh,b^a=bcefh,c^a=ce,d^a=dfi,e^a=cfh,h^a=fi,i^a=hi,c^b=cf,d^b=dg,e^c=ef60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083594);