Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C2 |
| direct product | SL(2,𝔽3)×#97 |
| unambiguous semidirect product | (Q8×#97)⋊C3 |
| Fitting, Sylow | Q8×#97 ⋊ C3 |
314 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 256 |
140 conjugacy classes — view the full character table.
| maximal subgroup of | #97×PSU(3,𝔽2)⋊C3, #97×GL(2,𝔽3) |
| maximal normal subgroup of | #97×GL(2,𝔽3) |
| maximal quotient of | #97×PSU(3,𝔽2)⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083831);