Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#2143 ⋊ C2 |
| Sylow | C3 ⋊ #13395 |
422 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 89 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|c^2=dg,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,i^b=i^2,d^c=dh,e^c=egh,g^c=gh,i^c=i^272 conjugacy classes — view the full character table.
| maximal subgroup of | #72270128, #72270141, #72270154, … |
| maximal normal subgroup of | #72270128, #72270141, #72270154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 88827);