Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #12366 |
364 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 67 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=f,d^2=f,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=egh,g^a=gh,i^a=i^2,d^b=df,e^b=eg,g^b=gh,d^c=dg,e^c=eg,f^c=fh,g^c=gh57 conjugacy classes — view the full character table.
| maximal subgroup of | #72214691, #72215215, #72215227, … |
| maximal normal subgroup of | #72215215, #72215227, D3×#12366 |
| maximal quotient of | #72214691, #72226166 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 83337);