Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #17439⋊C3 |
| Fitting, Sylow | #17439 ⋊ C3 |
311 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
3,2^8|b^2=g,c^2=h,f^2=h,g^2=i,b^a=bdfi,c^a=cdfi,d^a=defhi,e^a=dh,c^b=cf,f^b=fh,f^c=fh,e^d=eh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083746);