Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C24 |
| direct product | C8×D4×A4 |
| unambiguous semidirect product | (C22×C8×D4)⋊C3 |
| Fitting, Sylow | C22×C8×D4 ⋊ C3 |
839 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 839 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | C8×D4×S4, A4×#1580, #72342696 |
| maximal normal subgroup of | C8×D4×S4, A4×#1580, #72342696 |
| maximal quotient of | A4×#1580, A4×#2171 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084633);