Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #7438 |
174 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=d,b^2=eg,c^2=fg,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,i^a=i^2,c^b=cf,d^b=defgh,e^b=eh,g^b=gh,d^c=dg,e^c=eh,e^d=eh57 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 57183);