Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C2×C3×#365 ⋊ C2 |
| Sylow | C3 ⋊ #13249 |
444 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 111 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=f,b^2=dfg,c^2=e,d^2=g,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,i^b=i^2,f^c=fh66 conjugacy classes — view the full character table.
| maximal subgroup of | #72225869, #72258562, D3×#13249 |
| maximal normal subgroup of | #72225869, #72258562, D3×#13249 |
| maximal quotient of | #72225874, #72226137 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 88016);