Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#1360 ⋊ C2 |
| Sylow | C3 ⋊ #11924 |
414 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|a^2=e,c^2=h,d^2=gh,e^2=fg,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,i^c=i^287 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 80790);