Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #33300⋊C3 |
| Fitting, Sylow | #33300 ⋊ C3 |
339 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 339 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|c^2=f,d^2=g,e^2=h,f^2=i,b^a=bhi,c^a=cdfgi,d^a=eg,e^a=defhi,f^a=fgi,g^a=h,h^a=ghi,c^b=cf,d^b=dg,e^b=eh,f^b=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72225964, #72228327, #72228328, … |
| maximal normal subgroup of | #72225964, #72228327, #72228328, … |
| maximal quotient of | C2×#33300⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084025);