Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #33381⋊C3 |
| Fitting, Sylow | #33381 ⋊ C3 |
286 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 45 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:
3,2^8|b^2=g,f^2=hi,h^2=i,b^a=bdgh,c^a=cdgh,d^a=deh,e^a=dg,c^b=cf,f^b=fh,h^b=hi,f^c=fh,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084057);