Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | C4⋊#4626 |
233 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 30 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:
2^4,3,2^4|a^2=d,b^2=c,f^2=h,g^2=i,b^a=bc,e^a=e^2,f^a=fg,g^a=ghi,h^a=hi,f^d=fh,g^d=gi,f^e=gh,g^e=fg,h^e=i,i^e=hi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085914);