Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#2165 ⋊ C2 |
| Sylow | C3 ⋊ #11992 |
412 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 79 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:
2^8,3|a^2=e,d^2=g,e^2=fg,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,i^b=i^2,e^c=eh60 conjugacy classes — view the full character table.
| maximal subgroup of | #72214823, #72215218, #72215232, … |
| maximal normal subgroup of | #72215218, #72215232, D3×#11992 |
| maximal quotient of | #72214823 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 81098);