Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #13269 |
308 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 107 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=e,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cg,e^b=eh,f^b=fh,i^b=i^2,d^c=dh,i^c=i^290 conjugacy classes — view the full character table.
| maximal subgroup of | #72270348, D3×#13269, #72270380, … |
| maximal normal subgroup of | #72270348, D3×#13269, #72270380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 88111);