Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #32881⋊C3 |
| Fitting, Sylow | #32881 ⋊ C3 |
1067 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 1067 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|b^2=i,f^2=i,b^a=bg,c^a=ch,d^a=egh,e^a=deh,g^a=h,h^a=gh,c^b=cf,d^b=dg,e^b=eh,f^b=fi,f^c=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72233784, SD16×#120⋊C3, #72241506, … |
| maximal normal subgroup of | #72233784, SD16×#120⋊C3, #72241506, … |
| maximal quotient of | C2×#32881⋊C3, #72341912, #72341916 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083974);