Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#1425 ⋊ C2 |
| Sylow | C3 ⋊ #13656 |
184 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=e,c^2=e,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ch,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh,i^b=i^2,i^c=i^290 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#13656, #72255019, #72255025 |
| maximal normal subgroup of | D3×#13656, #72255019, #72255025 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 90180);