Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #15190⋊C3 |
| Fitting, Sylow | #15190 ⋊ C3 |
433 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 433 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
3,2^8|b^2=i,c^2=f,d^2=g,e^2=i,b^a=bdfg,c^a=defgi,d^a=cdefghi,e^a=efgi,f^a=gi,g^a=fghi,c^b=cf,d^b=dg,e^b=ei,d^c=dh48 conjugacy classes — view the full character table.
| maximal subgroup of | #72278669 |
| maximal quotient of | #72299986 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083683);