Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#1366 ⋊ C2 |
| Sylow | C3 ⋊ #11989 |
292 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 69 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,c^2=g,d^2=g,e^2=fg,f^2=h,b^a=bd,c^a=cf,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,i^b=i^2,d^c=dh,i^c=i^257 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 81084);