Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C4⋊(A4⋊C16) |
221 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 221 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3,2^2|a^2=d,b^2=c,d^2=e,e^2=f,b^a=bc,g^a=g^2,h^a=hi,h^g=i,i^g=hi112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085909);