Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #13291 |
306 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 111 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=def,c^2=eg,d^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^2,c^b=cg,e^b=eh,f^b=fh,f^c=fh78 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#13291, #72270509, #72270547, … |
| maximal normal subgroup of | D3×#13291, #72270509, #72270547, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 88206);