Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#2122 ⋊ C2 |
| Sylow | C3 ⋊ #13103 |
646 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 646 conjugacy classes of subgroups — too many to draw. Only its 89 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:
2^8,3|a^2=f,b^2=dfh,c^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,e^c=eh,f^c=fh,i^c=i^2,f^d=fh57 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 87173);