Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C1 |
| direct product | SL(2,𝔽3)×M32 |
| unambiguous semidirect product | (Q8×M32)⋊C3 |
| Fitting, Sylow | Q8×M32 ⋊ C3 |
185 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
140 conjugacy classes — view the full character table.
| maximal subgroup of | M32×PSU(3,𝔽2)⋊C3 |
| maximal quotient of | M32×PSU(3,𝔽2)⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084080);