Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#1867 ⋊ C2 |
| Sylow | C3 ⋊ #12966 |
580 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 580 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|a^2=f,b^2=f,e^2=h,b^a=bd,c^a=ce,e^a=eh,g^a=gh,c^b=ch,e^b=egh,d^c=dg,e^c=eh,i^c=i^2,e^d=eh66 conjugacy classes — view the full character table.
| maximal subgroup of | #72270482, #72270490, #72270502, … |
| maximal normal subgroup of | #72270482, #72270490, #72270502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 86370);