Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #14899⋊C3 |
| Fitting, Sylow | #14899 ⋊ C3 |
291 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=fg,c^2=fg,d^2=f,e^2=h,b^a=bcdfh,c^a=cdhi,d^a=cfh,e^a=efgh,f^a=fg,g^a=fh,c^b=cf,d^b=dg,e^b=ei,d^c=dh48 conjugacy classes — view the full character table.
| maximal subgroup of | #72222703, #72222814, #72222832 |
| maximal normal subgroup of | #72222703, #72222814, #72222832 |
| maximal quotient of | #72222696 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083669);