Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#1953 ⋊ C2 |
| Sylow | C3 ⋊ #13417 |
454 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 454 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) = 96 |
Polycyclic presentation:
2^8,3|c^2=d,d^2=h,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,f^a=fg,g^a=gh,i^a=i^2,c^b=cf,d^b=dh,e^b=eg,f^b=fgh,g^b=gh,i^b=i^2,e^c=egh,f^c=fg,g^c=gh66 conjugacy classes — view the full character table.
| maximal subgroup of | #72271484, #72271490, #72271510, … |
| maximal normal subgroup of | #72271484, #72271490, #72271510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 88972);