Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #12780 |
756 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|a^2=f,c^2=h,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^2,d^b=dh,e^b=eg,f^b=fgh,d^c=dgh,e^c=eh,f^c=fh,e^d=eh,f^d=fh72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#12780, #72247796, #72247954 |
| maximal normal subgroup of | D3×#12780, #72247796, #72247954 |
| maximal quotient of | #72247980 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 85472);