Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#1304 ⋊ C2 |
| Sylow | C3 ⋊ #9495 |
1212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 97 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=df,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg,e^b=eh,f^b=fg,i^b=i^2,d^c=dh,f^c=fh,i^c=i^263 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#9495, #72231807, #72231812, … |
| maximal normal subgroup of | D3×#9495, #72231807, #72231812, … |
| maximal quotient of | #72231783, #72231786, #72231791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 67466);