Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#1353 ⋊ C2 |
| Sylow | C3 ⋊ #11906 |
440 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 77 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:
2^8,3|a^2=e,b^2=f,c^2=g,d^2=gh,e^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cfh,d^b=dg,e^b=efgh,d^c=dh,i^c=i^2,e^d=eh60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 80724);