Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #11669 |
360 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 95 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,c^2=f,d^2=fh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,i^a=i^2,c^b=cf,d^b=df,f^b=fh,d^c=dh,e^c=eh90 conjugacy classes — view the full character table.
| maximal subgroup of | #72214587, #72215218, #72215227, … |
| maximal normal subgroup of | #72215218, #72215227, D3×#11669 |
| maximal quotient of | #72214587, #72226145 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 79648);