Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #22536⋊C3 |
| Fitting, Sylow | #22536 ⋊ C3 |
339 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=f,d^2=f,e^2=f,h^2=i,b^a=bcefi,c^a=cef,e^a=cfi,c^b=cf,d^b=dg,g^b=gi,e^c=ef,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083879);