Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #34182⋊C3 |
| Fitting, Sylow | #34182 ⋊ C3 |
748 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 748 conjugacy classes of subgroups — too many to draw. Only its 133 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=i,c^2=g,d^2=h,e^2=g,b^a=befhi,c^a=efgh,e^a=cei,c^b=cg,d^b=dh,f^b=fg,e^c=eg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084183);