Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #34196⋊C3 |
| Fitting, Sylow | #34196 ⋊ C3 |
650 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 117 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,e^2=g,f^2=gh,b^a=bcg,c^a=e,e^a=ce,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, 1084188);