Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #62510⋊C3 |
| Fitting, Sylow | #62510 ⋊ C3 |
435 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 69 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=gi,d^2=h,f^2=g,h^2=i,b^a=bcei,c^a=ei,e^a=cef,c^b=cg,d^b=dh,f^b=fg,h^b=hi,d^c=di,e^c=eg,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084921);