Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C22×C3×AGL(1,ℤ/8) ⋊ C2 |
| Sylow | C3 ⋊ #23439 |
2026 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,g^b=gh,i^b=i^2,e^c=eh,f^c=fh,g^c=gh90 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#23439, #72248415, #72253287, … |
| maximal normal subgroup of | D3×#23439, #72248415, #72253287, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 196780);