Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C23×M16 ⋊ D3 |
847 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 847 conjugacy classes of subgroups — too many to draw. Only its 79 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:
2^6,3,2^2|c^2=d,d^2=f,e^2=f,c^a=ce,e^a=ef,g^a=g^2,h^a=hi,c^b=cf,e^c=ef,h^g=i,i^g=hi64 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#421, #72341544 |
| maximal normal subgroup of | S4×#421 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087090);