Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C3×#1389 ⋊ C2 |
| Sylow | C3 ⋊ #12649 |
428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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=e,b^2=f,c^2=e,d^2=g,f^2=gh,g^2=h,b^a=bd,d^a=dg,f^a=fg,g^a=gh,c^b=cg,e^b=eh,i^b=i^2,d^c=dh,f^c=fh,i^c=i^278 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#12649, #72269947, #72269978, … |
| maximal normal subgroup of | D3×#12649, #72269947, #72269978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 84853);