Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C3×#585 ⋊ C2 |
| Sylow | C3 ⋊ #32899 |
3866 conjugacy classes of subgroups; only the 315 normal subgroups are shown.
This group has 3866 conjugacy classes of subgroups — too many to draw. Only its 315 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|d^2=h,b^a=be,c^a=cf,d^a=dg,i^a=i^2,f^b=fh,e^c=eh117 conjugacy classes — view the full character table.
| maximal subgroup of | #72271194, #72271216, D3×#32899, … |
| maximal normal subgroup of | #72271194, #72271216, D3×#32899, … |
| maximal quotient of | #72271944 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 318344);