Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #13345 |
772 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 772 conjugacy classes of subgroups — too many to draw. Only its 81 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=h,c^2=h,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,g^b=gh,d^c=dg,i^c=i^2,f^d=fh63 conjugacy classes — view the full character table.
| maximal subgroup of | #72248015, D3×#13345, #72248157 |
| maximal normal subgroup of | #72248015, D3×#13345, #72248157 |
| maximal quotient of | #72271610 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 88517);